Category Archives: Number Theory & Primes

Activities with Number Theory and Primes

RSP Palindromes

     I like to play chess on the internet. It is often the case that players are rated with numbers according to how well they perform. Recently I noticed an interesting bit of number trivia about my rating in a certain type of chess. It said that I had 1661 points! (Not bad, see but not the best of the players.)

     Of course, try I was happy, because it was a palindrome. But upon looking more closely, it can be observed that 16 is a square number, and its reverse, 61. is a prime number! Moreover, this is unique for all squares from 1 to 100.

     So what do we have here? Well, WTM wants to call something like this a Reversible-Square-to-Prime Palindrome, or RSP Palindrome, for short.

     Here is a chart of all numbers less than 100 (with one exception) that produce RSP Pals.

n Square Prime Palindrome Prime Factors
4 16 61 1661 11 x 151
14 196 691 196691 11 x 17881
19 361 163 361163 11 x 32833
28 784 487 784487 11 x 71317
32 1024 4201 10244201 11 x 127 x 7333
37 1369 9631 13699631 11 x 1245421
38 1444 4441 14444441 11 x 17 x 77243
41 1681 1861 16811861 113 x 17 x 743
62 3844 4483 3844483 7 x 112 x 45389
85 7225 5227 72255227 11 x 6568657
89 7921 1297 79211297 11 x 127 x 56701
95 9025 5209 90255209 11 x 79 x 283 x 367
97 9409 9049 94099049 11 x 232 x 103 x 157

     Now, dear reader, you are invited to continue this list. Send any results you find and WTM will post them here.

     As to the exception referred to above… 402 = 1600. The reverse of 1600 is either 0061, or 61 if the leading zeros are suppressed. This gives us 16000061 and 160061 as two more RSP Pals for this range.

     Next, the curious thinker should be asking himself… what about cubes and their reversals? Do similar cases exist for RCP Palindromes? The answer is not long in coming to light. Observe:

53 = 125      521 is prime.      Hence 125521 is a RCP Pal.

     Except for the trivial 503 case, how many RCP Palindromes can be found for n < 100?


Trotter’s Own Prime Oddities

For the present, this page is only meant to be a storage place for curious and odd facts that WTM has discovered while researching prime numbers and the prime factorizations of various composite numbers. The dates indicate when the item was submitted for posting in the website Prime Curios.


  • 11: Begin with 11, and continually [i.e. recursively] add the first five powers of 2, but in reverse order (32, 16, …, 2). All sums are primes (43, 59, 67, 71, and 73). Sent 8/8/01
  • 41: The sums of the squares of the first digits with the cubes of the second digits of the primes in the first prime triplet (41, 43, 47) — i.e. ab gives a2 + b3 — are primes as well (17, 43, 359). [Note: 43 produces itself.] sent 8/7/01
  • 164: Its prime factorization is 2 x 2 x 41. Then 12 + 62 + 4(4-1) = 101, a pal-prime. Then changing (4-1) to (4×1) and (4+1) produces two more primes, 293 and 1061, respectively. Sent 8/6/01
  • 168: A factorization of 168 is 3 x 7 x 8. So, 13 + 67 + 88 = 17057153, which is prime. [Note: 17, 05, 71, and 53 are primes as well.] sent 8/7/01
  • 263:Cloning the digits of this prime as exponents in this way — 22 + 66 + 33 — yields another prime: 56687. sent 8/4/01
  • 323 Patrick De Geest has said: “323 doubled up (i.e., 323323) has five consecutive prime factors which when squared and summed, yield 989, another palindrome!” WTM adds: And when those prime factors (of 323323) are merely summed, or cubed before summing, a prime number is the total each time (67 and 15643, respectively). [Note: 15643 happens to be part of a twin prime pair and of a prime quadruple.]
  • 463: Cloning the digits of this prime as exponents in this way — 44 + 66 + 33 — yields a composite (46939), which upon deleting the 9’s, leaves 463. sent 8/7/01
  • 643: Cloning the digits of this prime as exponents in this way — 66 + 44 + 33 — yields a multiple of itself: 46939 (= 73 x 643). Sent 8/4/01
  • 881: Cloning the digits of this prime as exponents in this way – 88 + 88 + 11 — yields a rather interesting result: 33554433, whose prime factorization is 3 x 11 x 251 x 4051. (Note the lengths of the 4 primes). Sent 8/4
  • 881: [2nd var.] Using the clones of the digits of this prime, in reverse manner – 81 + 81 + 18 – yields the prime 17. 8/7/01
  • 997: Cloning the digits of this prime (the largest 3-digit prime) as exponents in this way – 99 + 99 + 77 — yields another prime: 775664521. sent 8/4
  • 997: The largest 3-digit prime AND the sum of the squares of its digits is also a prime (211). sent 8/4

Palindromes and Prime Factorizations

Observe these interesting patterns. Let’s begin with this basic palindrome: 98789. It’s main claim to fame is that it’s the largest 5-digit palindrome that is the sum of three consecutive primes. (Can you find those primes?)
The palindrome is not a prime because its prime factors are 223 x 443. But now notice this: 223 + 443 = 666, the number of the Beast!
We’re not through with this. Watch! Let’s “squeeze” the digits of 98789 from the sides until the “7” disappears and the “8’s” merge into one. This gives us another, smaller palindrome: 989. Still not a prime, but check out its prime factorization and the sum of those factors:

23 x 43 and 23 + 43 = 66.
(Might we not consider 66 as a “baby” beast?)

It looks like we’re on to something here. Let’s continue with a larger palindrome: 9876789. Its prime factorization is

9876789 = 33 x 13 x 19 x 1481

No “beastly” number here, you say. Ah, but look closely as we re-arrange those prime factors a little…

(3 x 3 x 13 x 19) x (3 x 1481)

which yields the following…

2223 x 4443 and 2223 + 4443 =6666

…and it looks as though our beast is growing up!

There are certainly more palindromes to investigate. Try these. Your task is to re-arrange the primes to produce a pair of numbers that has a sum of “all 6’s”. Sometimes it’s easier than others.

987656789 = 72 x 71 x 313 x 907

98765456789 = 61 x 3643 x 444443

9876543456789 = 34 x 17 x 97 x 1697 x 43573

987654323456789 = 172 x 29 x 5303 x 22222223

98765432123456789 = 449 x 494927 x 444444443

9876543210123456789 = 32 x 13 x 6353 x 8969 x 1481481481

tt(8/26/01)

Postscript (9/3/01): Another presentation of the concept above can be found in Patrick De Geest’s The World of Numbers, as WON plate 112.


A Closer Look at Another Pattern

In the work above, we highlighted 2 numbers in blue: 1481 and 1481481481. The reason, of course, is that there is something special about them in addition to being primes. The second number shows why: it is composed of a block of digits “148”, repeated three times, then it ends with a “1”.
That should make you wonder about 1481481. It is easily seen that it is not prime — the sum of its digits is a multiple of 3 — so it must have a prime factorization. If you divide it by 3, then 3 again, then by the largest 2-digit prime, you will see a nice result.
Now we have three numbers that form a “family”: 1481, 1481481, and 1481481481. And two of those were prime.
I’ll bet you know what the next question will be, right? Naturally, what happens if we use more blocks of “148”? It should be obvious that our numbers become rather large; so we feel it’s time for a little new notation. We will illustrate our method with the 3rd number: 1481481481.
It has three blocks of “148”. We will show this as (148)3. So with the final “1”, our number looks like this:

(148)3 … 1.

In general, we denote our numbers in this way:

(148)k … 1, where k = 1, 2, 3, 4, …

It just so happens that we have checked the values of k up to 14. Here is what we found:

(148)k … 1 is prime for k = 1, 3, and 4.

Can anybody go further?


Mirror, Mirror, On the Wall

Let’s now turn our attention to the “mirror images” of our numbers. Reversing 1481 gives us 1841. But while 1481 is prime, its reversal is not. Proof: 1841 = 7 x 263. You see, changing the positions of the “8” and “4” made a big difference.

Does reversing digits in 1481481 make any difference? That is to say, could its reverse (1841841) now be prime? Unfortunately, the answer is NO. The reason is that changing digit-order does not change the sum of the digits of the number. It is still a multiple of 3. (Can you find its prime factorization?)

However, for (148)3 … 1, change does have a big effect. Now (184)3 … 1 is composite. Here is a partial factorization. Can you finish it?

1841841841 = a2 x b x 196799

Are you ready for a big surprise now? Here ’tis…

(148)4 … 1 and (184)4 … 1

are both PRIME!

Continuing with this theme, we can now state: (184)k … 1 is composite for k = 5 to 11. Beyond that is unexplored territory.


Sandwich Primes

After further thought, WTM has decided to call any prime that starts with the digit “1”, and ends with the digit “1”, as a sandwich prime.

Our first such prime occurred in the palindrome investigation above: 1481. The extreme digits, the 1’s, serve as the “slices of bread”, and any other digits represent the fillings.

And if we continue repeating the block of digits as shown in other numbers above, we have a refinement in our new name: Dagwood primes! (Recall the famous character in the comics, Dagwood Bumstead, who often made multi-layer sandwiches with extra slices of bread separating his fillings.)

So our first Dagwood prime to be offered is this: 1481481481.

Our investigation of sandwich primes has turned up some interesting results, which we will share with you now.


We begin by noting that our research of 1481 was inspired by the factorization of the palindrome 9876789, and then the factor 4443. The factors of 4443 are 3 and 1481. So it seemed a natural extension to examine the number 5553.

Step 1: 5553 = 3 x 1851. But 1851 is not prime; it is 3 x 617.

Step 2: Let’s repeat it in this manner: 1851851. Bingo! A sandwich prime, of the Dagwood variety!

Step 3: And repeating again — 1851851851 — yields an even bigger prime!

Step 4: Unfortunately, further repetitions, up to k = 12, yield no more primes.

We may summarize the foregoing this way: (185)k … 1 is prime for k = 2, 3.

tt(9/4/01)

Reversing the digits in this manner gives this:

(158)k … 1 is prime for k = 2, 12.

Wow! Look at that last value for k. That’s special. Here it is, in full glory:

1581581581581581581581581581581581581581

Here is a table, summarizing all the data gathered to date (k < 13):

Number Form Primes when k =
(148)k … 1 1, 3, 4
(184)k … 1 4
(158)k … 1 2, 12
(185)k … 1 2, 3
(123)k … 1 1, 2
(132)k … 1 1, 6, 10
(147)k … 1 1, 7
(174)k … 1 1, 2
(138)k … 1 1, 2
(183)k … 1 1, 2, 3, 4, 6, 11
(115)k … 1 1,
(151)k … 1 1, 3, 4, 6
(102)k … 1 1, 4, 5
(120)k … 1 1, 2, 3, 7, 12
(103)k … 1 1,
(130)k … 1 1, 3
(106)k … 1 1,
(160)k … 1 1,
(109)k … 1 1, 4, 12
(190)k … 1 1, 7

Trotter in Prime Curios

Background

In the latter part of May of this year (2001) we discovered a very interesting website all about prime numbers, titled appropriately The Prime Pages. There is a companion page connected with it, called Prime Curios, a collection of clever and interesting trivia, moderated by G. L. Honaker, Jr. It is to this 2nd site that this WTM page is concerned.

Naturally, we began submitting our own contributions right away. First, we sent the one about 1992 that appears at the beginning. Then others began to follow in rapid succession. The list soon grew rather lengthy, so we decided to place ours in one location. So what you will see and read below is the results of our number play since that time. Nothing is in any particular order, unless otherwise indicated.

Each entry in the list is preceded by a link (the highlighted number) that will take you to the specific page in the website of Prime Curios, so that you may read all the other interesting facts that other people found about that particular number. We think you will be quite surprised and well rewarded.

So go forward now, and most of all, have fun with numbers!


  1. 1992 1992 = 8 x 3 x 83. The only other two years in the period 1000 to 1999 to share the structure of “a x b x ‘ab'” (where ‘ab’ is prime and the concatenation of the factors a and b) are 1316 = 4 x 7 x 47 and 1533 = 7 x 3 x 73.
  2. 1992 If you separate the digits of 1992 like 199 2, you have two primes. Note that 199 is the largest prime less than 2 hundred. Separation in the middle gives us 19 92, and 19 turned upside down along with 92 reversed are prime.
  3. 2310 2310 = 2 x 3 x 5 x 7 x 11 = 112 + 132 + 172 + 192 + 232 + 292. Note the consecutive digits: 0, 1, 2 and 3.
  4. 197 197 is the only three-digit prime Keith number.
  5. 26 The prime factorization of 26 uses the first three counting numbers.
  6. 17 17 is the smallest Trotter prime, i.e., a prime of form 10 x (n2) + 7, where n = 1, 2, 3 …
  7. 1234567 The prime factors of 1234567 (127 x 9721) form a peak-palindromic arrangement of digits (1279721). It is curious that the prime factors of 1279721 are all three emirps (79 x 97 x 167) which upon concatenation form yet another prime (7997167). [Note: see this reference…World of Numbers for more information.]
  8. 36 The smallest square that is the sum of a twin prime pair: 17 and 19.
  9. 8 8 is the smallest cube which is the sum of a twin prime pair {3 + 5}.
  10. 23 2n + 3n is prime for n = 0, 1 and 2.
  11. 23 23 is the smallest prime for which the sum of the squares of its digits is also an odd prime.
  12. 31 The number of letters (in English) required to write the word names of the first six primes is the reverse of the sixth prime (13), namely 31.
  13. 73 73 is the smallest prime whose square (5329) is the concatenation of two multi-digit primes.
  14. 73 389 and 17 are primes, as is their concatenation (38917). Inserting the lowly 0 between them transforms the prime into a power, i.e. 389017, the cube of 73, a prime itself.
  15. 83 The cube of 83 (571787) is the smallest case of the concatenation of a pair of 3-digit primes.
  16. 211151 The smallest Xmas Tree Prime with 3 rows, i.e. 6 digits.
  17. 2111511013 The smallest Xmas Tree Prime of 4 rows, i.e. of 10 digits.
  18. 211151101310867 The smallest Xmas Tree Prime of 5 rows, i.e. of 15 digits.
  19. 3883 This palindrome is transformed into a pal-prime upon the insertion of a single 0 in the middle.
  20. 121661 121661 is the smallest OP-PO Prime.
  21. 72727 72727 is a pal-prime, and separating the digits so — 72 and 727 — we can now say the sum of the digits of the first 72 primes (2, …, 359) is 727, another pal-prime. (P.S. I’m submitting this on 7/27 of this year!)
  22. 277 76729 is the square of the prime 277 and the smallest square with 5 or more digits that is the concatenation of three primes (7, 67, and 29). (Note: the square contains the digits of the root embedded in reverse order.)
  23. 159 159 = 3 x 53, and upon concatinating the prime factors, we have a peak palindrome, 353, which is itself a prime.
  24. 159 Its square (25281) is the concatenation of 2 primes: 2 and 5281.
  25. 77 The square of 77 is 5929, the concatenation of two primes, 59 and 29.
  26. 181 181, a pal-prime, is the sum of the digits of the first 23 primes (2, …, 83).
  27. 217 47 and 89 are primes, as is their concatenation (4789). Inserting the lowly 0 between them transforms the prime into a power, i.e. 47089, the square of 217.
  28. 311 311 is the 11th three-digit prime for which the sum of the squares of its digits is also a prime; and the sum here is 11 as well.
  29. 2357 21 + 33 + 55 + 77 and 22 + 33 + 55 + 77 are twin primes. [Trotter, Kulsha]
  30. 2357 Letting A=1, B=2, …, Z=26, then 2357 is the sum of all the values of the U.S. Presidents’ last names from Washington to Coolidge. [Ref. Wordsworth.]
  31. 2357 2357 is also the sum of consecutive primes in at least two ways: (773 + 787 + 797) and (461 + 463 + 467 + 479 + 487).
  32. 17 Using A = 1, B = 2,…, Z = 26, 17 is the smallest non-negative number whose numerical value of its word form is also a prime (109).
  33. 7 Using A = 1, B = 2, …, Z = 26, the sum of all the letter values in the word names of the numbers from 1 to 7 is a prime (367). If 0 is included, the sum is yet another prime (431).
  34. 64 Using A = 1, B = 2, …, Z = 26, the sum of all the letter values in the word names of the numbers from 1 to 64 is a prime (7369). If 0 (whose alpha-numeric value is 64) is included, the sum is yet another prime (7433).
  35. 5 Only 5 U.S. Presidents have 5 letters in their last names (Both Adams, Grant, Hayes, and Nixon).
  36. 23 23 = 14 + 23 + 32 + 41 + 50.
  37. 13 Using the first three primes we have: 23 + 5 = 13.
  38. 111 111 equals the sum of 2 + 3 + 4 + … + 17 minus the sum of the primes less than 17.
  39. 131 131, a palindromic prime, equals the sum of 2 + 3 + 4 + … + 19 minus the sum of the primes less than 19.
  40. 414 The exponential factored form of 414 (2 x 32 x 23) consists of three 2’s and two 3’s; whereas its expanded form (2 x 3 x 3 x 23) has two 2’s and three 3’s.
  41. 434 434 is also the sum of the cubes of the digits of the emirp 347.
  42. 821 The smallest prime of the first prime quadruple for which the sums of the cubes of the digits of the 4 primes (821, 823, 827, 829) are primes themselves (521, 547, 863, 1249).
  43. 440 The sum of the first 17 primes (2 to 59) and also the number of yards in the quarter-mile race in track-and-field competitions.
  44. 997 The largest 3-digit prime AND the sum of the cubes of its digits is also a prime (1801).
  45. 137 The sum of the squares of the digits of 137 is 59, another prime, and all five odd digits are used (Ref. Father Primes).
  46. 317 The sum of the squares of the digits of the prime 317 is 59, another prime. Note that all odd digits are present.
  47. 165 165 is a multiple of (16 – 5), which is its largest prime factor.
  48. 132 The concatenation of its three distinct prime factors (2, 3, and 11) forms primes in three ways: 2311, 2113, and 1123.
  49. 131143 This prime is composed of three 2-digit primes — 13, 11, and 43.
  50. 123456789 Replacing each of the digits, one-by-one with a 0, yields primes in three cases: 1, 2, and 7 (023456789, 103456789, and 123456089). Note that 127 is a Mersenne prime.
  51. 37 37 + 4n yields primes for n = 1, 2, 3, 4, 5, 6, 7.
  52. 45 If the first five powers of 2 (2, 4, 8, 16 & 32) are each subtracted from 45 all results are primes (43, 41, 37, 29 & 13).
  53. 170 The 170th Trotter number (289007) has an all-emirp prime factorization: 37 x 73 x 107. Note: the 3rd prime is a permutation of the digits of the original number.
  54. 304589267 A prime containing 9 distinct digits, where upon inserting symbols [30/45 + 89/267], we discover the missing digit “1”. The only other prime for which this is possible is 536948207. [Trotter and Knop]
  55. 13831 13831 is the smallest multi-digit palindromic prime such that the sum of it with the next prime (13841) is a palindrome (27672).
  56. 10501 A palindromic prime that is the sum of 3 consecutive primes (3491, 3499, and 3511), while at the same time serving as the middle prime of a set of three consecutive primes whose sum is another palindromic prime (31513).
  57. 97679 96769 is the largest 5-digit palindromic prime that is the sum of 3 consecutive primes (32251 + 32257 + 32261).
  58. 94949 The only 5-digit palindromic prime that is undulating and the sum of 3 consecutive primes (31643 + 31649 + 31657). Note that by adding 3 consecutive primes we only get one other undulating palindrome (16161), which is a non-prime.
  59. 98789 The largest 5-digit palindrome that is the sum of 3 consecutive primes (32917 + 32933 + 32939). Its prime factorization is 223 x 443 and 223 + 443 = 666!
  60. 13124…97909 (24-digits) A prime composed of eight 3-digit palindromes of a “consecutive” style, and one-nineth of (118)8 … 1.
  61. 11111117 11111117 and 71111111 are both primes, thus emirps.
  62. 742950290870000078092059247 (27-digits) The first prime in an arithmetic sequence of 10 palindromic primes. It was found by Dubner and his assistants and has common difference of 10101 x 1011.
  63. 144169 It is also the concatenation of three squares (144, 16, and 9). Note that: sqrt (144) = sqrt (16) x sqrt (9). [Note: this is an extension to another person’s “curio”.]
  64. 8609The largest distinct-digit pime. Pimes (pronounced with a long i) are primes whose digits contain circles, i.e., using only the digits 0, 6, 8, 9. Note: 6089 and 8069 are also distinct-digit pimes.
  65. 174 174 = 72 + 53 (using the first four primes).
  66. 199 199 is also a Permutable prime, meaning that 919 and 991 are primes as well.
  67. 2213 2213 is a “sum of cubes” as follows: 23 + 23 + 133.
  68. 2222 The smallest number divisible by a 1-digit prime, a 2-digit prime, and a 3-digit prime.
  69. 1429 The prime sum of two famous baseball records: 714, number of homeruns hit by Babe Ruth, and 715, number of the homerun hit by Hank Aaron to break the Babe’s record (on 4/8/1974).
  70. 202 A semiprime palindrome equal to (2 + 3 + 5 + 7)2 – (22 + 32 + 52 + 72). It’s the only such case for all primes < 2,000,000,000. [Trotter and De Geest]
  71. 576 A square equal to (2 + 3 + 5 + 7 + 11)2 – (22 + 32 + 52 + 72 + 112). It’s the only such case for all primes < 2,000,000,000. [Trotter and De Geest]
  72. 223 The sums of the nth powers of its digits are prime for all n between 1 and 6 inclusive: sum of digits = 7, sum of squares of digits = 17, sum of cubes of digits = 43, sum of fourth powers = 113, sum of fifth powers = 307 and sum of sixth powers = 857.
  73. 607565706 A palindrome resulting from this “prime based” expression: (257 + … + 607)2 + (2572 + … + 6072). There are 57 consecutive primes inside each parentheses. Note that this palindrome starts with the last prime added: {607}565706. “57” appears also as a substring, 60756{57}06. The number of the beast {6}075{6}570{6} is included. [De Geest and Trotter]
  74. 107 Rudy Giuliani was the 107th mayor of New York City.
  75. 2003 There is only one way to use consecutive integers to produce a sum of 2003.
  76. 20022002 The prime factorization of 20022002 is 2 x 7 x 11 x 13 x 73 x 137, which when grouped thus, 2 x 11, 13 x 73, and 7 x 137, yield 3 palindromic semi-primes: 22, 949, and 959.
  77. 1951 1951 is prime, and appears as the 9th term of the sequence 1+9+5 = 15, 9+5+15 = 29, 5+15+29 = 49, etc.
  78. 98689 The first centered triangular number (i.e. of the form the form of (3n2 – 3n + 2)/2) that is a palindromic prime.

This next group of 7 items is the result of some email correspondence we had with Carlos Rivera (6/27/01). [See Potpourri for that email.] We posed the basic idea, and Carlos provided us with the numbers. (See above on 36 and 8 for the cases of square and cube.)

  1. 253124999 The smaller of the smallest twin prime pair for which the sum is a 4th power (sum = 1504).
  2. 4076863487 The smaller of the smallest twin prime pair for which the sum is a 5th power (sum = 965).
  3. 578415690713087 The smaller of the smallest twin prime pair for which the sum is a 6th power (sum = 3246).
  4. 139967 The smaller of the smallest twin prime pair for which the sum is a 7th power (sum = 67). (Note: this prime ends with a 6 and 7.)
  5. 14097…72287 (26-digits) The smaller of the smallest twin prime pair for which the sum is an 8th power (sum = 15188). [The entire number is 140975 6730907423 9886172287.]
  6. 73099303486215558911 The smaller of the smallest twin prime pair for which the sum is a 9th power (sum = 1749).
  7. 8954942912818222989311 The smaller of the smallest twin prime pair for which the sum is a 10th power (sum = 16810).

The next items are the results of looking at someone else’s “curio”, then expanding a little on it. We suggest that you go to the Prime Curio page to see the full number and the name of the original submitter.

  1. 15555…55551 (33-digits) The digital sum of this prime is 157, another prime (whose digit sum in turn is yet another prime: 13).
  2. 10220…02201 (55-digits) The digit sum of this prime is 110, which is the double of its number of digits.
  3. 14444…44441 (67-digits) The digit sum of this prime is 262, a peak palindrome.
  4. 18181…18181 (77-digits) The digit sum of this 77-digit prime is 343, the cube of 7.
  5. 31313…31313 (83-digits) The digit sum of this prime is the prime 167.
  6. 19999…99991 (87-digits) The digit sum of this prime is a palindrome, 767.
  7. 37777…77773 (87-digits) The digit sum of this prime is 601, another prime.


Finally, we present some items that are harder to categorize. The first one was created by the moderator of Prime Curios, G. L. Honaker, Jr., after we wrote our webpage on Trotter Numbers and Trotter Primes. The second one is a two-person contribution, involving Monte Zerger (whose name and creations can be found in other WTM pages) and ourselves (WTM).

  • 735 There are exactly 735 Trotter primes less than 100,000,000. Note the first three odd primes in 735. [Honaker]
  • 510 The concatenation of 510 with itself (510510) is the product of the first 7 primes and also the product of the 7th through 10th Fibonacci numbers (13, 21, 34, and 55). [Zerger](Continuing the previous curio) The difference between the next prime (19) and the next Fibonacci number (89 – also a prime) is 70, which is the product of the Fibonacci subscripts above. [Trotter]
Note: Some of the above items have been removed from the Prime Curios page, though they at one time were indeed posted. Still it doesn’t alter the basic facts about any given entry. It was merely a decision taken later by the site moderator.

Father Primes

FATHER PRIMES

A “father” prime shall be defined as one for which the sum of the squares of its digits is also a prime. The sum is therefore the “child” prime.
Example: 23 is a “father” prime because 22 + 32 = 4 + 9 = 13. That is, 23 is the “father, (or progenitor)” of a prime child, namely 13. But, 13 is not a prime who can be a father (i.e. have a child), because 12 + 32 = 1 + 9 = 10, a composite number.
An example of an ancestral line of fathers and children might be this:

191 to 83 to 73.

191 is father to 83 and grandfather to 73; 83 is father to 73; but 73 is “childless”, as the sum (58) of the squares of its two digits (7 and 3) is a composite number. Hence, it is “the end of the line”.
The longest ancestral lines so far established have 5 generations. One example of these is

1499 to 179 to 131 to 11 to 2.

Problem: Find a “father” and “grandfather” for 1499.

Working Notes: (July 2001)

All two-digit fathers are given here.

11: 12 + 12 = 1 + 1 = 2 prime

23: 22 + 32 = 4 + 9 = 13 prime

41: 42 + 12 = 16 + 1 = 17 prime

61: 62 + 12 = 36 + 1 = 37 prime

83: 82 + 32 = 64 + 9 = 73 prime

% % %

list of father-child primes: 100 < Prime fathers < 1000.

prime child no. of fathers prime fathers
11 3 113, 131, 311
17 2 223, 401
19 2 313, 331
37 1 601
41 1 443
43 1 353
53 2 461, 641
59 3 137, 173, 317
61 2 463, 643
67 3 337, 373. 733
73 1 661
83 2 191, 911
89 1 229
97 1 409
101 2 467, 647
107 1 773
109 2 683, 863
113 1 449
131 4 179, 197, 719, 971
137 1 883
139 4 379, 397, 739, 937
149 1 829
163 3 199, 919, 991
179 2 797, 977
211 1 997
Total 47 .

113: 12 + 12 + 32 = 1 + 1 + 9 = 11 prime

131: same result.

311: emirp for 113; is the 11th term in the list of odd prime sums

137: 12 + 32 + 72 = 1 + 9 + 49 = 59 prime & all odd digits

173: same
317: same

371: 7 x 53 [but 33 + 73 + 13 = 27 + 343 + 1 = 371]

713: 23 x 31

731: 17 x 43

337: 32 + 32 + 72 = 9 + 9 + 49 = 67

373: same

733: same

179: 12 + 72 + 92 = 1 + 49 + 81 = 131

197: same

719: same

791: 7 x 113

917: 7 x 131

971: same

[note: 449 -(ssd)-> 113, a permutation of 131.]

379: 32 + 72 + 92 = 9 + 49 + 81 = 139

397: same

739: same

793: 13 x 61

937: same

973: 7 x 139 [7’s co-factor is the “ssd” sum of this group]

199: 12 + 92 + 92 = 1 + 81 + 81 = 163

919: same

991: same; emirp for 199.

Decade trios:

461 yields 53 641 yields 53

463 yields 61 643 yields 61

467 yields 101 647 yields 101

Strings:

179 gvs 131 gvs 11 gvs 2

191 gvs 83 gvs 73

463 gvs 61 gvs 37

443 gvs 41 gvs 17

111611 [or 611111] gvs 41 gvs 17

22441 [or 24421, or 44221] gvs 41 gvs 17

449 to 131 to 11 to 2

449 div by 81 = 5 r 44

so five 9’s, and 6, 2, & 2 could be used.

22699999/no

26299999/no

62299999/yes, a prime

So 62299999 to 449 to 131 to 11 to 2 is another ancestral family tree of five generations.

1699 gvs 199 gvs 163

35466227 gvs 179 gvs 131 gvs 11 gvs 2 another family tree of 5 generations.

1499 gvs 179 gvs 131 gvs 11 gvs 2; which is the one given at the top.

1499 — > 1 + 16 + 81 + 81 = 179

179 — > 1 + 49 + 81 = 131

131 — > 1 + 9 + 1 = 11

11 — > 1 + 1 = 2

1499 prime 4199 no 9149 no 9419 prime
1949 prime 4919 prime 9194 no 9491 prime
1994 no 4991 no 9914 no 9941 prime

Trotter Numbers & Trotter Primes

Recently (June 2001) I became aware of an interesting website, dedicated to the discovery and reporting of appearances of the number 47 in our world. It is called, appropriately enough, the 47 Society. They post e-mail notes from the members about any trivia related to what they claim is the “quintessential random number”. Well, if you have read the pages of WTM, it should come as no surprise to you that I “enlisted” in the society. And on June 8, I wrote my first e-mail to them, which said:

Hey, I like your neat project about 47. While I’m not quite ready to believe that 47 is the only number worth looking for, :>) , I do enjoy looking for number facts of any kind. So here’s my humble contribution…

About 9 years ago I wrote a letter that was published in the Nov. ’92 issue of the MATHEMATICS TEACHER (NCTM) about “1992”. You see, 1992 = 8 x 3 x 83. But also in the past thousand years only 2 other years had that same structure: a x b x ‘ab’. They were 1533 = 7 x 3 x 73 and (ta-dah!) 1316 = 4 x 7 x 47.

This morning as I lay in bed thinking about “47” (yes, this is true), it struck me that “47” was the concatenation of “4” [a square; I like squares, too] and “7” [the “ubiquitous 7″, as I like to say]. So I began examining other such cases.

We get 17, 47, 97, 167, 257, 367, … all primes so far. [But of course, 497 isn’t prime, but that was sorta to be expected.] Some future terms from here on are primes, while others are not.

BUT 47 is the 2nd prime in this sequence, and 2 is the only even prime. So that might count for something, huh? [Which brings to mind this quote: All primes are odd except 2–which is therefore the oddest of them all. [Knuth] ]

I hope you like this, and I’ll keep my eye out for more 47’s, okay?


That little comment about the sequence of numbers containing the number 47 was the inspiration of all that follows in this article. I warn you — it gets wild at times. Enjoy.

Definition

The set of Trotter Numbers is a subset of the natural numbers, or positive integers, defined by the following rule:

T(n) = 10 * n2 + 7, where n = 1, 2, 3, …

The sequence begins: 17, 47, 97, 167, 257, 367, 497, …

Whenever a given T(n) [aka TN] is prime, it shall be called a Trotter Prime (TP).

After a few moments of close observation and reflection, one should notice that while the first six consecutive TN’s given are prime, the 7th one, 497, is a composite number. It is equal to 7 × 71. This characteristic alone, that in the TN sequence — unlike the sequence of natural numbers — primes can be consecutive, makes the set of Trotter Numbers interesting.

However, while it is quite easy to expect that among the infinite number of TN’s, some will be prime and some will be composite, you still have to test each TN to see if it is also a TP.

Lucky for us: we can easily prove that every 7th one can not be prime. Hence, there can never be a string of TP’s longer than 6. The question merely remains: do strings of 5, 4, or 3 TP’s exist? If so, where are they? That is what you, as the great prime hunter, must do — find ’em, and tag ’em!


This is where we stop on this page. Now it’s up to you. You must start investigating the topic of Trotter Numbers and Trotter Primes before you turn the page, as it were, to see what patterns or odd tid-bits of trivia you can find, then compare your results with what WTM has discovered so far.

When you finish with your reseach, collect all your notes together, and turn to Page 2. Thank you and good luck!

P.S. WTM is rather pleased to state that the sequence given above can be found in Sloane’s On-line Encyclopedia of Integer Sequences and has its own reference number: A061722.


Polygonal Numbers

A polygonal number is defined as “A type of figurate number which is a generalization of triangular, square, etc., numbers to an arbitrary n-gonal number. The above diagrams graphically illustrate the process by which the polygonal numbers are built up.” (Mathworld.wolfram.com) Every student of school mathematics knows about the square numbers, and many know about the triangular numbers as well. But less familiar are the pentagonal, hexagonal, etc. varieties.

Even less well known is the fact that each of those types of numbers has a cousin of sorts, called the centered polygonal numbers. Yes, the regular triangular numbers have their corresponding “centered” form. The same is true for the squares, pentagonals, hexagonals, etc. (See diagram below.)

Therefore, our definition for these numbers is “A figurate number in which layers of polygons are drawn centered about a point instead of with the point at a vertex.” (Mathworld.wolfram.com)

Many facts and theorems are known about polygonal numbers, especially of the squares and triangulars. We wouldn’t be able to talk about the Pythagorean Theorem if it weren’t for the squares, just to mention the most famous example of all. And the triangulars arise whenever we are concerned with the sum of consecutive integers, from 1 to n.

What I want to do in this page of WTM is present some ideas that are not normally covered in an average school math class, yet ideas that are well within the understanding of most students. First, we will show the algebraic formulas for both the regular and centered polygonal numbers, up to a level seldom discussed: a 30-sided polygon!

The Formulas
Number
of Sides
Regular
form
Centered
form
3 n(n + 1)/2 (3n2 – 3n + 2)/2
4 n2 2n2 – 2n + 1
5 n(3n – 1)/2 (5n2 – 5n + 2)/2
6 n(2n – 1) 3n2 – 3n + 1
7 n(5n – 3)/2 (7n2 – 7n + 2)/2
8 n(3n – 2) 4n2 – 4n + 1
9 n(7n – 5)/2 (9n2 – 9n + 2)/2
10 n(4n – 3) 5n2 – 5n + 1
11 n(9n – 7)/2 (11n2 – 11n + 2)/2
12 n(5n – 4) 6n2 – 6n + 1
13 n(11n – 9)/2 (13n2 – 13n + 2)/2
14 n(6n – 5) 7n2 – 7n + 1
15 n(13n – 11)/2 (15n2 – 15n + 2)/2
16 n(7n – 6) 8n2 – 8n + 1
17 n(15n – 13)/2 (17n2 – 17n + 2)/2
18 n(8n – 7) 9n2 – 9n + 1
19 n(17n – 15)/2 (19n2 – 19n + 2)/2
20 n(9n – 8) 10n2 – 10n + 1
21 n(19n – 17)/2 (21n2 – 21n + 2)/2
22 n(10n – 9) 11n2 – 11n + 1
23 n(21n – 19)/2 (23n2 – 23n + 2)/2
24 n(11n – 10) 12n2 – 12n + 1
25 n(23n – 21)/2 (25n2 – 25n + 2)/2
26 n(12n – 11) 13n2 – 13n + 1
27 n(25n – 23)/2 (27n2 – 27n + 2)/2
28 n(13n – 12)/2 14n2 – 14n + 1
29 n(27n – 25)/2 (29n2 – 29n + 2)/2
30 n(14n – 13) 15n2 – 15n + 1

Hey! Do you see a pattern in the table? If you do, perhaps you could write a general formula for it; then you could give the formula for any n-gonal number of either type, without using the table, and even beyond 30 sides.


And now for some numbers…

Our next chart will give us some actual numbers for the polygons up to decagons.

The First Nine Terms
Name Regular Centered
triangular 1, 3, 6, 10, 15, 21, 28, 36, 45, … 1, 4, 10, 19, 31, 46, 64, 85, 109, …
square 1, 4, 9, 16, 25, 36, 49, 64, 81, … 1, 5, 13, 25, 41, 61, 85, 113, 145, …
pentagonal 1, 5, 12, 22, 35, 51, 70, 92, 117, … 1, 6, 16, 31, 51, 76, 106, 141, 181, …
hexagonal 1, 6, 15, 28, 45, 66, 91, 120, 153, … 1, 7, 19, 37, 61, 91, 127, 169, 217, …
heptagonal 1, 7, 18, 34, 55, 81, 112, 148, 189 1, 8, 22, 43, 71, 106, 148, 197, 253, …
octagonal 1, 8, 21, 40, 65, 96, 133, 176, 225, … 1, 9, 25, 49, 81, 121, 169, 225, 289, …
nonagonal 1, 9, 24, 46, 75, 111, 154, 204, 261, … 1, 10, 28, 55, 91, 136, 190, 253, 325, …
decagonal 1, 10, 27, 52, 85, 126, 175, 232, 297, … 1, 11, 31, 60, 101, 151, 211, 281, 361, …

Now that we have some numbers, what should we do with them? If I may paraphrase an old popular song by Nancy Sinatra, these numbers are made for adding! So consider this…

We again turn to Mathworld for some vital information: Fermat’s Polygonal Number Theorem
In 1638, Fermat proposed that every positive integer is a sum of at most three triangular numbers, four square numbers, five pentagonal numbers, and n n-polygonal numbers. Fermat claimed to have a proof of this result, although Fermat’s proof has never been found. Gauss proved the triangular case, and noted the event in his diary on July 10, 1796, with the notation

What that little cryptic notation means is that all whole numbers can be expressed as the sum of three, or fewer, triangular numbers. Here is an interesting example:

100 = 91 + 6 + 3 = T13 + T3 + T2

100 = 55 + 45 = T10 + T9

This illustrates that sometimes a number has two possibilities, with 3 or 2 terms. Nice, huh?


Turning now to the case of the squares… Fermat said that all whole numbers can be expressed as the sum of four, or fewer, square numbers. Let’s look at this example:

50 = 49 + 1 = S7 + S1

50 = 25 + 25 = S5 + S5

50 = 25 + 16 + 9 = S5 + S4 + S3

50 = 36 + 9 + 4 + 1 = S6 + S3 + S2 + S1

Notice that there were expressions with 2, 3, and 4 terms. Thereby, arises an interesting idea: given a particular number, how many different expressions can be found? I challenge you to research this and report back to me. Ok?


One more time… For the case of the pentagonals, we can use up to five of them to express all whole numbers. Let’s check out the situation for the number 2002.

2002 = 1001 + 1001 = P26 + P26

2002 = 1520 + 477 + 5 = P32 + P18 + P2

2002 = 1820 + 176 + 5 + 1 = P35 + P11 + P2 + P1

2002 = 1717 + 176 + 92 + 12 + 5 = P34 + P11 + P8 + P3 + P2

As before, we have demonstrated that we can achieve our goal with 2-5 terms. In fact, there are many more such ways to do it than presented here. Finding all possible ways is now more difficult, (unless one uses a computer program).


The Other Side of the Story

So far we have only been working with the regular polygonal numbers. Let’s now look at the centered case. The natural question to ask should be: does there exist an analogue to Fermat’s theorem, as discussed above? Specifically, are three CTN’s (Centered Triangular Numbers) sufficient to express all whole numbers?

The best way to answer that is to start small and work your way up. Here is a chart for the numbers from 1 to 10. Recall, the set of CTN’s is {1, 4, 10, 19, …}.

The CTN Analogue
No. expression No. expression
1 1 6 4 + 1 + 1
2 1 + 1 7 none
3 1 + 1 + 1 8 4 + 4
4 4 9 4 + 4 + 1
5 4 + 1 10 10

Well, I guess that about answers our question, doesn’t it? And it didn’t take long either.

However, it brings to mind yet another question — what is the next impossible number?

And the next? And the next?

Then what happens when this idea is extended to CSN’s (Centered Square Numbers) and CPN’s (Centered Pentagonal Numbers)? What are the impossible values when using these other sets of numbers? And beyond?

[Remember: you can use up to 4 CSN’s and 5 CPN’s, and so on, in the expressons.]


Special Numbers

Another popular activity when one is faced with a long list of numbers is to search for the presence of numbers with special characteristics, such as squares, cubes, or palindromes. Let’s first consider the modern favorite of many mathematicians: palindromes.

The “mother of all websites” dealing with palindromes undoubtedly is World!Of Numbers, edited by Patrick De Geest. In his site you can find an extensive treatment of palindromes that are also triangular numbers, and squares as well. In fact, he gives data for the pentagonals up to the nonagonals. We heartily encourage you to visit that site; you will be justly rewarded for your time and efforts.

However, all the data to be found there uses only the regular type of polygonals; there is nothing mentioned about the centered type. Here is our attempt to fill in that gap of trivia information. (Note: At present, our search only shows results up to n = 300 and for k-gonals from k = 3 to 40. We also omit any single-digit palindromes as being trivial in this context.)

The Palindromes
k n CPkN(n) Prime Factorization
3 101 15151 109 x 139
174 45154 2 x 107 x 211
211 66466 2 x 167 x 199
249 92629 211 x 439
257 98689 prime
4 10 181 prime
13 313 prime
17 545 5 x 109
5 8 141 3 x 47
9 181 prime
65 10401 3 x 3467
6 18 919 prime
7 3 22 2 x 11
20 1331 113
8 6 121 112
51 10201 1012
56 12321 32 x 372
61 14641 114
9 4 55 5 x 11
12 595 5 x 7 x 17
10 2 11 prime
5 101 prime
11 5 111 3 x 37
7 232 23 x 29
11 606 2 x 3 x 101
12 727 prime
62 20802 2 x 3 x 3467
12 5 121 112
6 181 prime
16 1441 11 x 131
46 12421 prime
13 5 131 prime
14 5 141 3 x 47
8 393 3 x 131
9 505 5 x 101
15 5 151 prime
10 676 22 x 132
52 19891 prime
16 5 161 7 x 23
46 16561 prime
17 5 171 32 x 19
93 72727 prime
18 3 55 5 x 11
5 181 prime
8 505 5 x 101
40 14041 19 x 739
19 5 191 prime
20 4 121 112
21 2 22 2 x 11
9 757 prime
36 13231 101 x 131
120 149941 11 x 43 x 317
255 680086 2 x 11 x 19 x 1627
22 none in this range yet
23 7 484 22 x 112
29 9339 3 x 11 x 283
24 7 505 5 x 101
25 4 151 prime
36 15751 19 x 829
289 1040401 101 x 10301
26 30 11311 prime
27 8 757 prime
28 35 16661 prime
29 3 88 23 x 11
30 4 181 prime
94 131131 7 x 11 x 13 x 131
260 1010101 73 x 101 x 137
31 69 72727 prime
32 2 33 3 x 11
10 1441 11 x 131
26 10401 19 x 739
251 1004001 3 x 334667
33 260 1111111 239 x 4649
34 25 10201 1012
43 30703 prime
172 500005 5 x 11 x 9091
35 25 10501 prime
28 13231 101 x 131
33 18481 prime
36 25 10801 7 x 1543
260 1212121 prime
37 none in this range yet
38 31 17671 41 x 431
39 52 51715 5 x 10343
260 1313131 17 x 77243
40 3 121 112
9 1441 11 x 131
27 14041 19 x 739
225 1008001 prime

While studying the results above, I saw two rather interesting numbers: 1212121 and 1313131. Not only do they share an obvious digital pattern of 1d1d1d1, but they both are the 260th term in their respective orders (k = 36 and 39).

Now if we look back at the 33-gonal and 30-gonal lists, we see 1111111 and 1010101. As one might begin to suspect by now, they are the 260th terms there. So it’s table time again!

The 260th Term Case
k Number Prime Factorization
30 1010101 73 x 101 x 137
33 1111111 239 x4649
36 1212121 prime
39 1313131 17 x 77243
42 1414141 43 x 32887
45 1515151 11 x 181 x 761
48 1616161 prime
51 1717171 199 x 8629
54 1818181 31 x 89 x 659
57 1919191 29 x 66179

Stop the Presses!!! (4/22/2002)

Just in to the editorial offices of WTM! Patrick De Geest has sent in a pair of 6-packs…of palindromes for the missing data in the chart of Palindromes above. Here it is.

Palindromes for k = 22 & 37
k n CPkN(n) Prime Factorization
22 4156 189949981 13 x 14611537
524962 3031430341303 7 x 13 x 33312421333
321895111 1139781083801879311 13 x 53 x 163 x 15259 x 665102447
358542860 1414082803082804141 7 x 19 x 67 x 349 x 2671 x 170235089
362349816 1444271276721724441 83 x 17400858755683427
422820435 1966548318138456691 17 x 191 x 5346613 x 113277481
37 378 2636362 2 x 163 x 8087
2400 106515601 43 x 2477107
407157 3066863686603 prime
2835585 148749979947841 859 x 56099 x 3086801
3443283 219339595933912 23 x 27417449491739
6792834 853637858736358 2 x 17 x 25106995845187

Also, my friend and colleague from Romania, Andrei Lazanu, sent along some important data regarding the matter above about the impossible cases for numbers to be expressed as the sum of 3, or fewer, CTN’s. According to the program he wrote, there are over 70 numbers less than 200 that can not be decomposed in this way. How many can you find? Can you find all of them?

Andrei also was kind enough to provide WTM with some information about how many ways 2002 can be expressed with 5, or fewer, regular pentagonal numbers. It can be done in 166 ways. [See examples above.]

In addition, he informs us that the same task can be achieved in 31 ways with regular triangular numbers, and in 101 ways using regular square numbers. (Thanks, Andrei.)


Update (5/13/02)

De Geest has provided WTM with some more CTN palindromes. Here they are (as of 4/25/02).

More CTN Palindromes
(n < 3,711,895,911)
n CTN Prime Factorization
920 1690961 29 * 58309
1258 3162613 101 * 173 * 181
1263 3187813 prime
1622 5258525 52 * 17 * 12373
1707 5824285 5 * 17 * 68521
170707 58281418285 5 * 39821 * 292717
904281 1635446445361 109 * 9461 * 1585889
1258183 3166046406613 173 * 13537 * 1351913
7901015 124852060258421 21589 * 5783133089
8659930 149988757889941 17 * 852013 * 10355321
12458598 310433303334013 101 * 3073597062713
17070707 582818040818285 5 * 89 * 461693 * 2836741
80472265 12951570707515921 13 * 17 * 113 * 1609 * 322326253
1616689804 5227371841481737225 52 * 941 * 104917 * 2117911937
1680689789 5649436330336349465 5 * 29 * 761 * 51197936746897
1705387644 5816694029204966185 5 * 1163338805840993237

Next (5/13/02), Andrei has extended the matter of Centered Hexagonal Numbers (CP6N) if ever so slightly…

More CP6N Palindromes
(n < 1000)
n CP6N Prime Factorization
601 1081801 7 * 154543
630 1188811 13 * 19 * 4813

Smith’s Problem

Recently (June 1999) an individual, by the name of Smith, sent in the following question to the MATH FORUM of Swarthmore College, in particular to the “Ask Dr. Math” feature of that website:

“If p(n) is the function which adds up all the divisors of a natural number n, then could you please list for me some numbers n which have the following properties:

  1. p(n) has exactly one factor of 2;
  2. p(n)/2 has fewer distinct prime divisors than n itself.”


That person was answered by “Doctor Pete”, who said:

Hi,

Using Mathematica, for all n < 10000,

            10     34     52     90    106    306    388
           468    490    810    850    954    976   1300
          1525   1666   1690   2650   2754   2890   3033
          3050   3492   3610   3626   4194   4212   4410
          5194   5200   5746   6066   6100   6292   6516
          6800   7290   7650   8586   8746   8784   9610
          9700

all satisfy both properties mentioned. Furthermore, these are the only such numbers in the range mentioned that satisfy both properties.For instance,

p(490) = 1026 = 2 × 33 × 19

whereas

490 = 2 × 5 × 72.

I don’t know of any formula for finding such numbers; there doesn’t appear to be a pattern in the above list. But who knows, there may be an expression for them. To be honest I haven’t really looked into them very deeply.

– Doctor Pete, The Math Forum


It seems rather interesting to WTM that, given such a rather simple pair of conditions, there should be so few numbers that meet them. And there is that one “black sheep” (3033), a true “odd-ball”, a “rose among the thorns” (or vice versa). Numbers can be very, well, strange sometimes, can’t they?

Ruth-Aaron Numbers

On April 8, 1974 when Hank Aaron surpassed the career home run record of 714 set by the immortal Babe Ruth by hitting his own 715th homer, mathematicians found yet another reason to cheer. You see, if we prime factorize these two numbers, we have the following:

714 = 2 × 3 × 7 × 17

and

715 = 5 × 11 × 13.

Now a quick examination reveals a unique property: the first seven primes, and only those, are used! That certainly wouldn’t happen very often, would it? But a deeper look produces another relationship:

the sums of the primes for each number are the same!

29 = 2 + 3 + 7 + 17

and

29 = 5 + 11 + 13

Surprising or spooky, take your pick. But if you are a true number lover, you should be asking yourself this question: “Are there other pairs of consecutive numbers that behave in this way?” According to Paul Hoffman’s book, “The Man Who Loved Only Numbers: The Story of Paul Erdös and the Search for Mathematical Truth”[1], Carl Pomerance did. And the answer was in the affirmative. However, as might be suspected, such uth-Aaron pairs, as they can be called, appear somewhat infrequently. He did a computer search of all the positive integers less than 20,000 and found only 26 pairs. The smallest pair is 5 and 6; the largest, 18,490 and 18491. [2]

Then, in the tradition of all good number theorists, he conjectured (but did not prove) that the number of such pairs was infinite. That is where Paul Erdös entered the picture, setting to rest the dilemma; again the answer was in the affirmative.


Where do we go from here? Is this just a little tidbit of mathematical whimsy, or can it be turned into something worthwhile, at least for students and teachers of school math? We believe it can, and here is how we do it.

Initially, the students must have a working background of two ideas: prime numbers and the prime factorization of numbers. Once that is established, the students can be presented with the topic of Ruth-Aaron numbers, starting off with the story about the home run records of the two famous baseball greats. This is followed up with mention of what Pomerance did to search for more instances. We usually tell the class that 5  and 6 is the smallest pair, showing why it is so. Then we suggest they start looking for more pairs themselves. At first, shock sets in; they don’t quite know what to make of it. It’s a good idea to assure them that the next two pairs are rather small.

After they have been found — (8, 9) and (15, 16) — we go for the really big show and give them the largest pair in Pomerance’s search, asking them to prove that it is in fact an R-A pair. With calculators at hand and basic knowledge of the divisibility tests normally taught in middle school math, the request is a reasonable one. All the primes necessary for the task are less than 50 and the sum of the prime sets is less than 100. ‘Nuff said.


Ted Alper, of the Education Program for Gifted Youth at Stanford University, has provided us with the complete list of R-A pairs for numbers less than 1,000,000. [3] By his count, there are 149 such pairs. Therefore, we can do many additional exercises in any of three ways: (1) present a specific pair and ask that it be proved by showing the prime factorizations and the equal sums; or (2) present a set of four or five number pairs (only some of which are legitimate cases) and say, “Which pairs given here are indeed R-A numbers?”, or (3) state that an R-A pair exists in a certain small range of numbers, say a decade, then ask the students to find which pair of consecutive numbers is the one we seek. An example of the last approach might be: find the R-A pair hidden in the range of numbers from 5400 to 5410.


But why let a good thing stop there? A good mathematician, like a good musician or artist, is always creating new variations on a basic theme. Here is one such variation among other possibilities: find a pair of consecutive numbers for which their RAV’s [i.e. Ruth-Aaron Value, or sum of the primes] are themselves consecutive. A simple example of this is 24 and 25. The RAV(24) = 2 + 2 + 2 + 3 = 9 and RAV(25) = 5 + 5 = 10. Many such pairs of this type exist, of course. According to Alper, there are 55 such pairs less than 100,000. But, there is a catch here. Those pairs are what we call “Type i“, where “i” is for “increasing”. This can easily be expressed using “function notation”, as follows:

RAV(n) + 1 = RAV(n+1)

This implies there must be a second category, “Type d“, where “d” is for “decreasing”. The pair of 14 and 15 serves as a simple illustration; RAV(14) = 9 and RAV(15) = 8. Again using functions, we have

RAV(n) – 1 = RAV(n+1)

Once again, Alper’s computer reveals that there are 51 such pairs less than 100,000 in this category. [4]

We believe the use of function notation provides a unique way to present this topic in an advanced math manner. And therefore justify its inclusion in school math lessons. It provides a new challenge to the students while preparing them for their future study of functions in algebra and beyond.

Additional ideas for investigation include the following:

(1) Do there exist pairs of consecutive numbers for which for some positive integer k > 1,

RAV(n) = kRAV(n+1)
or
RAV(n) = (1/k)RAV(n+1)?

[Put in middle school math terms, this merely says: “The RAV of one number is a multiple of the RAV of the other number.“] The answer is yes, as might be expected. There are 43 such pairs in the range of numbers less than 2000. And for the record, 222 pairs less than 20,000; 630 pairs less than 100,000; and 2927 pairs less than 1,000,000. [4]

(2) Do there exist numbers n such that n = kRAV(n) for some positive integer k > 1? This means the RAV of a number is a factor (or divisor) of the given number. An example: RAV(60) = 15 and 15 is a factor of 60. [Note: this is a trivial case when n is prime, because RAV(n) = n. So non-prime values of n are what we are seeking here.]

(3) Take a number and double it. Notice that the RAV of the doubled number is only 2 more than the RAV of the first number. Can you explain this? For example the RAV(12) = 7, but the double of 12 is 24 and the RAV(24) = 9, which is only 2 more.

(4) Observe that RAV(291) = 100, the “century” number. It is the smallest number with this property. What is the next largest number whose RAV is 100?

(5) Define a recursive relationship on the set of whole numbers as follows: For a given n, find RAV(n). Then find the RAV of that result. Continue this process of finding the RAV of each succeeding outcome, until… Well, what do you think will happen? We call this the longevity of the original number n.

An example is 91.  RAV(91) = 20.
                         RAV(20) = 9.
                               RAV(9) = 6.
                                    RAV(6) = 5.
                                         RAV(5) = 5  DEAD-END!

The process terminates with 5, because 5 is a prime.

This means 91 –> 20 –> 9 –> 6 –> 5. Therefore, the longevity of 91 is 4. [We might symbolize this by writing RAV4(91) = 5 and L(91) = 4.]

Numbers can now be categorized by (1) their longevity numbers, or (2) according to which primes they eventually arrive. [For a more in-depth discussion of this feature, see [5].]

It seems that the possible avenues for investigation are almost endless, or at least, certainly varied enough to encourage individual creativity by anyone willing to try something new.


References

[1] Hoffman, Paul. “The Man Who Loved Only Numbers: The Story of Paul Erdös and the Search for Mathematical Truth”. Hyperion: New York. 1998.

[2] Nelson, Carol, David E. Penney, and Carl Pomerance, “714 AND 715″. Journal of Recreational Mathematics, Vol. 7, No. 2 (1974), p. 87-89.

[3] Alper, Ted. E-mail, May 29, 1999.

[4] Alper, Ted. E-mail, June 10, 1999.

[5] Dane, Perry, “The ‘Prime Derivative'”, Journal of Recreational Mathematics, Vol. 7, No. 2 (1974), p. 111-115.


NOTES

For a different slant on home runs in baseball with mathematics, go to Mike Keith’s site on Maris-McGwire-Sosa Numbers.

For a discussion of Ruth-Aaron Triplets in The Prime Puzzles and Problems Connection, click HERE.

For more useful information about generating RA Pairs, go to Joe K. Crump’s Number Theory Web.


APPENDICES

Below are given the lists of the first ten cases for various categories as mentioned in the article. To see longer lists, click here.

1. The first ten cases of regular R-A pairs:

Pair 1: (5,6) with sum 5.
Pair 2: (8,9) with sum 6.
Pair 3: (15,16) with sum 8.
Pair 4: (77,78) with sum 18.
Pair 5: (125,126) with sum 15.
Pair 6: (714,715) with sum 29.
Pair 7: (948,949) with sum 86.
Pair 8: (1330,1331) with sum 33.
Pair 9: (1520,1521) with sum 32.
Pair 10: (1862,1863) with sum 35.

————————————–

2. Alper also considered the variation on the basic theme of merely finding consecutive pairs for which sums of the distinct primes in each prime factorization were used. Here are the first cases:

Pair 1: (5,6) with sum 5.
Pair 2: (24,25) with sum 5.
Pair 3: (49,50) with sum 7.
Pair 4: (77,78) with sum 18.
Pair 5: (104,105) with sum 15.
Pair 6: (153,154) with sum 20.
Pair 7: (369,370) with sum 44.
Pair 8: (492,493) with sum 46.
Pair 9: (714,715) with sum 29.
Pair 10: (1682,1683) with sum 31.

—————————

3. This list is for “Type i” pairs as explained in the article.

Pair 1: (2,3) with first sum 2.
Pair 2: (3,4) with first sum 3.
Pair 3: (4,5) with first sum 4.
Pair 4: (9,10) with first sum 6.
Pair 5: (20,21) with first sum 9.
Pair 6: (24,25) with first sum 9.
Pair 7: (98,99) with first sum 16.
Pair 8: (170,171) with first sum 24.
Pair 9: (1104,1105) with first sum 34.
Pair 10: (1274,1275) with first sum 29.

—————————-

4. This list is for “Type d” pairs as explained in the article.

Pair 1: (7,8) with sums (7,6).
Pair 2: (14,15) with sums (9,8).
Pair 3: (63,64) with sums (13,12).
Pair 4: (80,81) with sums (13,12).
Pair 5: (224,225) with sums (17,16).
Pair 6: (285,286) with sums (27,26).
Pair 7: (351,352) with sums (22,21).
Pair 8: (363,364) with sums (25,24).
Pair 9: (475,476) with sums (29,28).
Pair 10: (860,861) with sums (52,51).

————————–

5. List for RAVs in which one is a non-trivial multiple of the other.

Pair 1: (74,75) with sums (39,13).
Pair 2: (160,161) with sums (15,30).
Pair 3: (174,175) with sums (34,17).
Pair 4: (252,253) with sums (17,34).
Pair 5: (259,260) with sums (44,22).
Pair 6: (287,288) with sums (48,16).
Pair 7: (335,336) with sums (72,18).
Pair 8: (391,392) with sums (40,20).
Pair 9: (395,396) with sums (84,21).
Pair 10: (447,448) with sums (152,19).

——————–

6. List of the first cases where the RAV is a factor of the given number.

Value 1: 16 has RAV  8.
Value 2: 27 has RAV  9.
Value 3: 30 has RAV  10.
Value 4: 60 has RAV  12.
Value 5: 70 has RAV  14.
Value 6: 72 has RAV  12.
Value 7: 84 has RAV  14.
Value 8: 105 has RAV  15.
Value 9: 150 has RAV  15.
Value 10: 180 has RAV  15.

Niven Numbers

The topic of Niven Numbers is briefly mentioned on another page in this website. But since writing that, I have been working on the concept a little bit more with some students and I now feel that it deserves its own separate page. So what follows here is a discussion about how it can be utilized in the regular school math class.


We begin by defining once again the concept of what a Niven Number is:

A Niven Number is any whole number that is divisible by the sum of its digits.


The only real trouble with this is the meaning of the word divisible. In number theory it simply means “division with no remainder“. Another way to express that is to say that the divisor in the division process is also a factor of the dividend.

An example or two should make all that quite clear. Let’s take the number 126. First, we must find the sum of the digits.

1 + 2 + 6 = 9

Next, we try to divide 126 by 9. And we find…

126 ÷ 9 = 14

There it is. 14 and no remainder; or put another way: 9 is a factor of 126 because

126 = 9 × 14

So 126 is a NIVEN NUMBER.

On the other side of the coin 121 is not a Niven Number. Its digit sum of 4 is not a divisor (or factor) of the number itself. 121 divided by 4 yields a quotient of 30 and a remainder of 1.

Doesn’t sound too hard now, does it? Well, it all depends. For young students this has proved to be sufficiently challenging, mostly because it is all so different from the usual math fare that is served up to them by their textbooks and teachers.

For example, I have posed the following simple question to students:

How many Niven Numbers can you find from 100 to 200?

It has been interesting to observe their initial reactions. Mainly they don’t quite know where to begin, what numbers to pick, and so on. They seem to be overly concerned about being unable to state the number of Niven Numbers in that range in a short amount of time, rather than realizing that it will take some time and patience to arrive at it. Next, they start out in an unorganized way, choosing numbers at random to test, rather than “beginning small and working up”.

So, after allowing them to struggle with the whole matter for a while, I make the suggestion, “Why not make a list of all the numbers from 100 to 200, inclusive; then start at the 100 and work your way up? We can circle the Niven Numbers, and cross off those that are not.” This shows a good problem solving strategy that many had never thought of. And at the same time, it allows us to review the divisibility tests that the students had studied earlier, but for lack of use in any meaningful context, had forgotten.

This organized strategy also has a unique advantage. One begins the find some easy Niven Numbers rather early: 100, 102,
108, 110, 111, and 112. Nothing breeds confidence and understanding better than some early success.

Furthermore, in this range of 13 numbers (100-112) there are several good opportunities for discussing another number property that aids our search. You see, there is no real need to carry out the long division and find the quotient IF the main objective is to determine if a number is Niven or not. If the number in question is odd and the digit sum is even, it is clear that the number can be crossed off the list. For example, 101. It is odd, but its digit sum is even. No odd number can be divided by an even number without leaving a remainder of some sort. Or to put it another
way:

No odd number is the product of an even number and something else.

So numbers like 101, 103, 105 and 107 can be eliminated rather easily. Another simple case arises when numbers have a digit sum of 10 and the number itself does not end with zero, as is the case for 109. Finally, in this limited range that we are considering, 104 is of interest. It has a digit sum of 5, yet the number does not end with a zero or 5. Basic and obvious as these points may seem to some, they still need to be brought into the discussion.

So with all these powerful ideas and strategies working for us now, just how many number from 100 to 200 are indeed Niven? We intend to leave that up to you to find. It’s not nice to spoil your “thrill of victory and agony of defeat”, right?


Where next?, you might be asking yourself now. Well, several ideas come to my mind; perhaps you could dream up some of your own. How about these two?

  1. What is the largest 3-digit Niven Number? (Also, the 2nd largest and 3rd largest ones?)
  2. What is the smallest Niven Number formed by the digits “1”, “6”, and as many zeros as needed?

Of course, we should mention the natural extension of the problem presented above: How many Niven Numbers are there in each “CENTURY” from 0 to 999? That is to say: 0-99, 100-199, 200-299, …, 900-999. The point here is to investigate these groups of one hundred numbers to see if there are any interesting patterns. Also to determine which century has the most (or the fewest) Niven Numbers. [Perhaps a bar graph could be the final presentation format?]

This would not be as tough a job as it might first appear if calculators, or even computers, were utilized. After all, it is the thinking, the mathematical investigation that is important here, not one’s ability at looonngg division. And don’t worry if your calculator doesn’t give division answers in “quotient-remainder” style. [Nowadays some do have that feature.] Any ordinary model will suffice, because of this basic fact:

If the division would produce a remainder in the traditional sense, then the calculator will give a “decimal” result instead.

Using our example earlier of 121, we would have this:

121 ÷ 4 = 30.25

And that is sufficient to tell us that there had to be a remainder of some sort, so 121 is therefore not Niven. (For the teacher and student willing to delve deeper into the matter, a discussion could be held about the relationships between ordinary remainders and the decimal part of the result. Thus making this project into a worthwhile learning experience.)

It’s now time to “get Niven”, and find those NN’s out there.

Good luck, and let me me know what you find.


POSTSCRIPT 1:

After writing the above article, I received an e-mail from the individual who first introduced me to this topic, a math professor in Colorado named Monte Zerger. Here I present selected portions of what he told me:

“You got me thinking about Niven numbers again, so I
did a little quick research. According to an article in the
Journal of Recreational Mathematics the origin of the name
is as follows. In 1977, Ivan Niven, a famous number theorist
presented a talk at a conference in which he mentioned
integers which are twice the sum of their digits. Then in
an article by Kennedy appearing in 1982, and in honor of
Niven, he christened numbers which are divisible by their
digital sum “Niven numbers.”

In most all the literature I have seen this is the accepted
name. However, I note that in his “Dictionary of Curious and
Interesting Numbers” Wells calls them “Harshad numbers.” …

In an article appearing in Journal of Recreational Mathematics in 1997, Sandro Boscaro defines and discusses Niven-morphic numbers. These are Niven numbers which terminate in their digital sum. For example 912 is such a number since it has a digital sum of 12, is divisible by 12, and ends in 12. In this article he proves a startling result. There is no Niven-morphic number which has a digital sum of 11, is divisible by 11 and ends in 11. 11 is the *only* such exception.

Of course 1, 2, 3, …,9 are all trivially Nivenmorphic. I played around this morning with a simple BASIC program and found the smallest (I think!) Nivenmorphics for 10, 12, 13, …, 19. They are [answers withheld to give you the chance to find them. Sorry, that’s the way things are in the World of Trotter Math. :=) ]”

POSTSCRIPT 2:

Later our correspondence grew a bit more, and the following curious trivia resulted:

  1. In my investigation for NNs under 1000 in order to understand what would be involved for the “century” question above, I found a string of 4 consecutive ones: 510, 511, 512, 513. And that was the longest string in that range.
  2. To which Monte replied, “My computer program found three other strings of four terms: 1014-1017, 2022-2025, and 3030-3033. I still like the string starting at 510 the best since 510 is the Dewey Decimal classification for mathematics and “cloning” it we have the product of the first 7 prime numbers as well as the product of the 7th through 10th Fibonacci numbers.”
  3. And Monte’s BASIC program uncovered the following item: “The smallest string of five consecutive Nivens begins with 131,052. I checked all the way up to 10,000,000 and could not find a string of six.”

    (Well, I think that’s going high enough for most purposes with young students in school. Perhaps a worthwhile activity would be to ask them to confirm that the numbers mentioned above are indeed Niven Numbers.)

    Then later Monte came through with a “really big” piece of information: “Ran across something today that will interest you. It has been proven there cannot be a string of more than 20 consecutive Nivens. A string of 20 has been found. Each member has 44,363,342,786 digits.”

  4. I also thought of the reverse side of things… How about the idea of strings of non-NNs? Between what two NNs can we find great numbers of non-NNs? In my limited, yet accessible-to-young-students list, I found these 17-term “droughts”: 558-576, 666-684, 736-756, 846-864, & 972-990. No Nivens here!
  5. Here’s more from the mind of Monte: “I suppose you noticed that whereas the current decade had only one Niven year (1998), the next decade has four! 2000, 2001, 2004 and 2007.”

[Stay tuned, dear reader. There’s no telling where this is going to end!]


Well, it took awhile, but there is some new action about Niven Numbers. On May 17, 2001, I received the following email:

I just noted your page on “Niven Numbers” and enjoyed it…………………I’m still very much interested in such numbers. Ivan Niven died last year but at least his numbers will help people to remember him. By the way, have you investigated Niven numbers in differenct bases? Have you considered which powers of two are Niven?

Sincerely,

Robert E. Kennedy

Ok! Different bases and powers of two… hmm… Looks like we have
more work to do.


GoldBach’s Conjecture

Here’s an “oldie but goodie” for you this time!

Some time long ago a mathematician, named C. Goldbach, was playing around with prime numbers and noted the following oddity:

All even numbers, greater than 2, can be expressed as the sum of two primes.

This is really a simple idea, as these few examples will show:

20 = 7 + 13 34 = 3 + 31 42 = 13 + 29

66 = 19 + 47 72 = 11 + 61 100 = 47 + 53

So what’s so hard about that, you ask? Well, just try proving that it is true for ALL even whole numbers! There’s the rub. Ever since Goldbach made this observation, mathematicians, professional and amateur alike, have tried to do that. But with no real success.

And that’s why it is called a “conjecture”, my friends.

A conjecture is a statement that you think is probably true, at least based on all the information at hand. But a “for sure” proof is not available. You see, it is not enough, as in a case like the one before us, to just say, “See! All the even numbers I’ve tested so far can all be expressed as the sum of a pair of primes. So all of them must behave the same way.” Mathematics doesn’t work that way.


So where do we go from here, you ask? Well, one of my favorite activities for my students is to ask the question:

How many pairs of primes can be found for any specific given even number?

This can be shown quickly by an example: 24.

5 + 19 7 + 17 11 + 13

So 24 can be expressed with three pairs of primes.

For those interested in making this investigation a bit more elegant, we can define this with function notation of advanced algebra in the following way: Let GN(n), the “Goldbach Number of n“, be defined as the number of pairs of primes the sums of which are the number n.

So for our example, we would write

GN(24) = 3.


Now, it’s easy to see that this is a good function, as every n (input) has a single GN(n) (output). But what is not so obvious is this fact: as the value of n increases, the same can not always be said for GN(n). Here’s a specific example of what I am saying here:

24 < 26 but GN(24) = GN(26)

because GN(26) also equals 3. [3+23, 7+19, and 13+13]

And even more significant is the case for n = 26 and n = 28. Observe:

26 < 28 but GN(26) > GN(28)

because GN(28) is only 2. [5+23 and 11+17] And there are many more such instances of this phenomenon. How many can you find?

Speaking in the language of algebra, we could say:

If n < n+k for some positive integer k, it is not always true that

GN(n) < GN(n+k).

Now here’s your assignment, class…

Find the GN values for all the even numbers from 4 to 100.

To assist you with your work here, a nice table of all the primes between 1 and 1000 can be found by clicking HERE.


Extension

Why stop with pairs of primes and the even numbers? Let’s also use trios
of primes and go for the odd numbers. Mr. Goldbach did, or so I hear.

You surely will find plenty to challenge you here.