# 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