Tag Archives: Primes

The Three Little Primes

You are about to read a special story problem, one I call a “mathematical fairy story” problem. You may find it a little hard at first, but that’s okay. If a story problem is easy for you to do, then it really isn’t a “problem” anymore, is it?”

Piet Hein, a Danish mathematician, architect, poet, philosopher, is famous for his little short poems, called Grooks, that contain little bits of wisdom. My favorite one says:

Worthy of attack, Prove their worth By hitting back.

So if my problem below “hits back”, then it has proved its worth.

Once upon a time there was a peripatetic philosopher, named Pieros Parios, who lived on his own little private island, called Utopia, just off the coast of Greece. The island was precisely pi kilometers from a prominent peninsula. But he had such a kind spirit and held no prejudice toward anyone, that the people who lived on this island often lovingly called him by the paternalistic appellation of “Papi Pieros”. Being the philanthropic man that he was, he established a pair of schools, one private and the other public, for which he served as the principal. He also founded several pristine hospitals, and many other such institutions that provided for the needs of the island’s plentiful inhabitants. However, there were no police or prisons on this island, because the population was so content and happiness reigned so completely that no one had any propensity to commit any act of impropriety.

Our story takes place in the middle school where Ms. Omega O is the math teacher. To bring a little diversion to her particular pedagogical style of instruction, she gives each pupil a new name at the start of each academic period. She first writes the students’ names in alphabetical order on a piece of paper. Then the first student on the list is renamed “Alpha A”. The second student becomes “Beta B”. The third is “Gamma G”, and so on, using the letters of the classical Greek alphabet. Everyone thinks it is quite a fun way to practice the Greek alphabet whose letters are so frequently used in higher level math books. One day Ms. Omega decided to begin her class with a problem related to a topic very dear to her heart: prime numbers. After taking the roll, she turned to the students and said, “I’m going to call on three of you to help me in the preparation of a little problem about prime numbers.” By the look of anticipation on the faces of the pupils as she spoke, she knew that she could predict that a successful learning experience was about to begin. So she continued: “Alpha, Beta, and Gamma, each of you please take out a small piece of paper and write on it any specific prime of your choice that is less than 100. Then bring your papers to my desk.” Gamma, who always liked to turn in his tests and quizzes prior to anyone else, promptly scribbled his selection and rushed forward to hand in his paper. Alpha and Beta, did not, on the other hand, share the same predilection for haste as Gamma. Being more prim and patient, they took a few more moments before they politely got up from their desks and handed in their papers. (After all, they were girls, who usually are predisposed to behave in a more appropriate manner than boys at this age.) Upon perusing the three papers, Ms. Omega’s face changed to a very pensive mood. Soon, with a smile appearing on her face, she began speaking to the class, saying: “I have now come to a decision. Here is your problem to solve. I’m sure I don’t have to remind any of you that all primes are positive whole numbers. Okay? So I will provide you with two other facts about the numbers that were picked by Alpha, Beta, and Gamma. I’ll use their last names, that is, A, B, and G, to state those facts. So watch carefully as I print them on the chalkboard. When you complete your solution, bring your paper to me and put it on my desk. Oh, and one more point. Calculators are not permitted on this problem.” She went to the chalkboard and wrote:
    1. A + B is a square number.

    2. (A – B) – G = 0.

Then she said, “The question that you need to answer is: What is product of the three primes?” Immediately, all the pupils began writing avidly with their pencils and ballpoint pens on their papers. But, almost instantly, Gamma, pushing his chair back from his desk, promptly rushed up to Ms. Omega with his paper, obviously full of pride that he was the first to be finished. A few moments later, Alpha, dressed in her cute blue pinafore, rose from her desk and gave her paper to the teacher. Beta did likewise just a few seconds later. The rest of the class, however, continued pondering the matter, with quite puzzled looks on their faces. Actually, Ms. Omega was not entirely surprised at this, because these three children were the most precocious in the class, in general. They usually scored quite well in math aptitude on standardized tests. She picked up Gamma’s paper first and began reading it. Although his penmanship wasn’t always the most legible of the class, she certainly could understand his basic premise. But as she began comparing it with the girls’ presentations, her eyes really began sparkling. Their work was easier to read, of course, but she then realized that something truly remarkable had transpired and she was experiencing one of those rare occasions that inspire teachers to call a “teachable moment”. (That’s when something unexpected happens that merits a second, more in-depth analysis of a situation.) And as the rest of the students began finishing the problem and turning in their papers, Ms. Omega soon had quite a pile of work on her desk. She then realized she had a lot of grading ahead of her, pouring over all the submissions. She might even have to cancel her appointment with her podiatrist that afternoon. Ah well, that’s the life of a dedicated professional pedagogue.

Well, that’s basically where our story ends. Now it’s your turn to do some serious puzzling and problem solving, by posing some questions for you to ponder:

  1. Can you find a logical explanation to account for the change of expressions on Ms. Omega’s face as she read the solutions submitted by the three children?
  2. Regarding the naming of the students, we will tell you that the ratio of male to female individuals present in the classroom during this perplexing incident was 1:1. (I guess it’s only fair to warn you that the teacher is one of the individuals being counted.) If Papi Pieros, in his role as school principal, were to assign one more new student to this particular section, Ms. Omega would have a little trouble assigning a special name to him/her. But since that hasn’t happened yet, tell us the number of boys in this group.
  3. Also regarding the naming system applied by the teacher, another strange oddity is evident. Some of the students’ names become palindromes when written in the style described in the story. How many such cases of palindromic names are there? List them.
  4. The late Paul Erdös, considered by many to be the most prolific writer of articles about mathematics of this century, perhaps of all time, never married, nor had a family of his own. Yet he dearly loved young children and they loved him. He used one of the Greek letters as his affectionate way to refer to them. What was that letter and why did Erdös undoubtedly choose that particular letter?
  5. This story was written with a special theme in mind. It will be rather subtle for some readers to deduce, or very mysterious, if you will, for others. And maybe impossible for the remaining people to figure out at all. But there are, we believe, plenty of clues scattered about to provide the necessary information for the most intrepid of problem solvers. Which pigeonhole do you fit in?
  6. The title of this story even has a familiar ring to it, doesn’t it seem to you? Could you draw a connection between the two that makes delicious sense in the culinary arts?

If your wish to send in solutions to some or all of the questions posed above, we encourage it.

Here are my two main addresses: trottermath@gmail.com or ttrotter3@yahoo.com.


The following day, after the pupils finished passing into the classroom and getting seated, who should enter but none other than Papi Parios himself! He often visited the classes in his schools, (partially to check up on his staff to see if they were promoting his particular philosophy – but he also enjoyed the experience of learning something new about math).

Paying no attention to his presence, Alpha and Beta raised their hands, requesting permission to speak. Ms. Omega turned to Alpha first, “Yes, Alpha, what’s on your mind today?”

“Well, Ms. O,” she said (using her teacher’s proper last name due to the fact that the principal was looking on), “after yesterday’s inspiring lesson, Beta and I have been playing around with our own little puzzle, using three primes. May we show it to you on the chalkboard?” “Of course, you may. Come forward, girls.”

Alpha took the chalk first, speaking as she wrote the numbers, “We have discovered something peculiar about the first three consecutive primes: 2, 3, and 5. If we multiply the larger pair, 3 x 5, we get 15. Now add 1 to obtain 16. Finally, we divide by the smallest prime, 2. And the answer, 8, is integral!’ (Alpha always loved to show off by using fancy, high level terminology.)

“Very nicely explained, Alpha,” said Ms. Omega. “But by the very basic principles of odd and even numbers, you can always expect that kind of result. Can you see why?”

“Yes,” Alpha replied. “We already thought of that. Here is our formula as proof.” Turning once again to the board, she wrote:

   odd x odd + 1        odd + 1          even
 -----------------  = ------------  =  -------- = integer   (Q.E.D.)
         2                 2              2

“Now that’s impressive, indeed!” beamed Ms. Omega. From the back of the room, Papi Parios began applauding enthusiastically.

“But that’s not all we discovered, Ms. O,” interjected Beta, with obvious pride in her voice. “Our pattern works for the next set of three consecutive primes where all the numbers are odd.” Without saying another word, she took the chalk from her companion and, stepping up to the board, wrote the following:

               5 x 7 + 1      35 + 1       36
              -----------  = --------  =  -----  =  12
                   3             3          3

“Great work, girls. You certainly are to be complimented, don’t we think so, everybody?” Principal Parios and the entire class, with one exception, nodded in agreement. Of course, that exception was Gamma, who had already gotten out his laptop computer, and was eagerly pressing his keyboard, entering data of some sort.

“But, alas, our pattern failed to work on the next set of three consecutive primes, namely, 5, 7, and 11,” Beta continued, with an obvious look of disappointment on her face. “Anyway, we spied another small bit of number trivia hidden in our two successes. The results of adding 1 to the products always yielded square numbers, 16 and 36.”

Sensing another “teachable moment” was at hand, Ms. Omega said to the whole class, “Perhaps if we were to take out our calculators, we could extend our search more efficiently and find another example of the pattern that Alpha and Beta have discovered for us. After all, it’s the search that’s important here, not the multiplication and division, right?”

Just then, Gamma began jumping up and down beside his desk, shouting excitedly, “Eureka! Eureka!” [That’s Greek for “I found it!”] “I’ve found another case of the pattern!”

Some kids turned to him and asked, “What is it, Gamma? What is it?”

Turning off his laptop and closing the cover, he smugly said, “Find it yourself! I’m keeping it a secret. It’s easy. At least, when you’re as smart as I am.”

Can you find Gamma’s solution? It really is easy. But I must let you in on a little secret… He has been continuing his search for a 4th solution for quite some time, with no luck. So it looks like a 4th solution just might be hard after all.


WTM would like to recognize two friends who helped make this page extra special. First, thanks go to Panagiotis Stefanides of Athens, Greece, who suggested real Greek names that I could use for the owner of the island. Second, we thank G. L. Honaker, Jr., from Bristol, VA. He created the idea of the three consecutive primes puzzle that was used in the Epilogue section. Without their kind contributions, this fairy story problem would not be as nice as it is (IMHO).

Number Sequences

[Preface NOTE: The material presented below was lifted from an article I saw some time ago by Dan Brutlag, “Making Your Own Rules”. THE MATHEMATICS TEACHER, November 1990. pp. 608-611. What is being given below is how I prepared a handout to give to my classes, mostly as anactivity for use by a substitute.]


Students: I found the information below in a math magazine. I think you will find it interesting, as I did.


  1. Study/investigate the results of applying the rules below to many different numbers. Keep good records to see if something “special” always happens. Describe what you find.
  2. Then make up a set of rules of your own to investigate, similar to or completely different from the samples given. Use the table of ideas that you find at the end of this handout to help you with this. Write up a little report about what you discover.

A. Lori’s Rule

  1. Start with any three-digit number.
  2. To get the hundreds digit of the next number in the sequence, take the starting number’s hundreds digit and double it. If the double is more than 9, then add the double’s digits together to get a one-digit number.
  3. Do the same thing to the tens and units digits of the starting number to obtain the tens and units digits of the new number.
  4. Repeat Steps 2 and 3 as often as necessary to find the special happening.

Example: 567, 135, 261, 432, ???, ???, …

B. Barbette’s Rule

  1. Start with any three-digit number.
  2. Add all the digits together and multiply by 2 to get the hundreds
    and tens digits of the next number in the sequence.
  3. To get the units digit of the next number, take the starting
    number’s tens and units digits and add them. If the sum is more than 9, add the digits of that sum to get a one-digit number for the units place.
  4. Repeat Steps 2 and 3 as often as necessary to find the special happening.

Example: 563, 289, 388, 387, ???, ???, …

C. Ramona’s Rule

  1. Start with any three-digit number.
  2. Obtain the next number in the sequence by moving the hundreds digit of the starting number ito the tens place of the next number, the tens digit of the original number into the units place of the next number, and the units digit into the hundreds place.
  3. Then add 2 to the units digit of the new number; however, if the sum is greater than 9, use only the units digit of the sum
  4. Repeat Steps 2 and 3 as often as necessary to find the special happening.

Example: 324, 434, 445, 546, ???, ???, …

D. Lisa’s Rule

  1. Start with any three-digit number.
  2. If the number is a multiple of 3, the divide it by 3 to get a new number.
  3. If it is not a multiple of 3 then get a new number by squaring the sum of the digits of the number.
  4. Repeat Steps 2 and 3 as often as necessary to find the special happening.Example: 315, 105, 35, 64, ???, ???, …

    Example: 723, 241, 49, 169, 256, ???, ???, …,


    Add ___ to ___ Divisible by ___
    Move to another position Hundreds, tens, or units place
    Take positive difference Prime
    Double Composite
    Square Even, Odd
    Halve (if even) Multiple of ___
    Multiply by ___ Greater than
    Replace by ___ Less than
    Exchange Equal to

    Postscript (7/29/99)

    Here is a personal sidelight to this activity. On April 23, 1991, I designed my own sequence rule-set. Here it is.


    1. Start with a four-digit number.
    2. To get the thousand’s digit and hundred’s digit of the next number, double the sum of the first three digits of the original number, that is, the thousand’s, hundred’s, and ten’s digits. (If this result is only a one-digit number, append a 0 to the left of that number, so a “6” would become “06”.)
    3. To get the ten’s and unit’s digit of the next number, double the sum of the last three digits of the original number, that is, the hundred’s, ten’s, and unit’s digits. (If this result is only a one-digit number, do as was done in Step #2.)

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


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.


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:


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


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


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


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.



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.


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.

Sum Primes

To view the Prime Chart, click HERE.

The concept of prime number is one that has fascinated mathematicians and frustrated students for a long time. It is one of the most fertile fields where one can make interesting discoveries on any level one wishes. By way of illustrating this, you only have to check out a few of the pages in this website: “Sexy” Primes, Primes & Square Pairs, Big Primes, and Paul Erdös. And a few more are in the planning stage.

I wish to turn my attention to something that is not so well known in the literature, but based on a math problem I once saw in a book. It concerned the sum of 3 primes that was itself a prime. The details are lost in my memory as to the exact numbers used, except to say that the problem, as stated, had an error in it. But it set me to thinking about the idea that will be explained here.

We begin in a time-honored fashion, with a couple of definitions:

Consecutive-Prime Sum (CPS): a number that is the sum
of two or more consecutive prime numbers.
Prime-Sum Prime (PSP): a prime number that is the sum
of two or more prime numbers.

A moment’s reflection on these definitions is necessary before proceeding further. A CPS is nothing more than a number that results from chosing any number of consecutive primes and finding their sum. That sum may take on various characteristics: prime or composite, naturally, or such things as squares, palindromes, etc. Whereas a PSP results from choosing any prime numbers one wishes (consecutive or not, with repetitions, etc.); it’s just that the sum itself must be prime.

Now, if we combine these two ideas into one, we really have something special:

a CPSP!!

Before we get lost in this alphabet soup, let’s look at a few examples:

CPS: 13 + 17 + 19 = 49
PSP: 11 + 13 + 19 = 43
CPSP: 11 + 13 + 17 = 41

Do you get it now? In the first case, the primes were consecutive; yet the sum did not happen to be a prime, in fact, it was a square number. In the second case, the primes were not consecutive, but the sum was a prime. But in the last case, the primes were consecutive AND the sum was also a prime. There we get the “best of both worlds”, as it were.

Well, now that that’s out of the way, where do we go from here? We have several options actually. One is to determine how many of the CPS‘s from 10 to 100 are themselves prime, that is, CPSP’s of order-3. In other words, limiting ourselves initially to three consecutive primes, how many solutions are there to this equation?

Pn + Pn+1 + Pn+2 = Pk

As it turns out, there are seven values for Pk, one of which was given above (41). We leave it to you to find the remaining six. Next, you can try four consecutive primes. Now you are looking for CPSP’s of order-4. [You should shortly make a very fundamental discovery when working on this case.] Later, you should pass on to 5 consecutive primes, etc.

A piece of advice is perhaps in order here: this now begins to be a good situation where you could use a spreadsheet to make the work more efficient.

There you have it! Now the rest is up to you. Dive in and see what you can discover.

To make all this prime hunting a little easier, we provide you with a chart of all the primes between 1 and 1000.

Big Primes

Preface: On this page I present a series of three handout sheets that I use in my Pre-algebra classes whose purpose is to teach how to find out if a large number is a prime or not. And by “large”, my main concern, as you will soon see, are numbers greater than 100 and less than 10,000. It is a bit lengthy, but it was designed to make certain the students have a firm grasp on the important background concepts necessary to deal with the work to be done in the whole activity. And it is presumed before seeing this material that the student already knows what a prime number is. My prefered definition is: a whole number, greater than one, whose only factors are itself and one.


Background: I

1. Square root: the square root of a number is a value which when multiplied by itself produces the given number. Examples:

4 is the square root of 16 because 4 × 4 = 16;
15 is the square root of 225 because 15 × 15 = 225; and
38 is the square root of 1444 because 38 × 38 = 1444.

2. Often a number does not have a whole number square root like the three examples above. For instance, there are no whole numbers such that when multiplied by themselves will give us 20, 85, or 150.

For 20, 4 × 4 = 16 is too small; and 5 × 5 = 25 is too big.

For 85, 9 × 9 = 81 is too small; and 10 × 10 = 100 is too big.

For 150, 12 × 12 = 144 is too small; and 13 × 13 = 169 is too big.

3. But by using decimals, we can get closer to the numbers. Observe:

For 20, 4.4 × 4.4 = 19.36 and 4.5 × 4.5 = 20.25 are better choices.

For 85, 9.2 × 9.2 = 84.64 and 9.3 × 9.3 = 86.49 are better choices.

For 150, 12.2 × 12.2 = 148.84 and 12.3 × 12.3 = 151.29 are better choices.

4. Our calculator can help us find these decimal values more rapidly than mere guessing. There is a key with a symbol like this: Pressing it after entering your number (on most calculator models) will give you an approximation of several digits.

For 20, we obtain 4.472135955.

For 85, we obtain 9.219544457.

For 150, we obtain 12.2474487.

Remember: these are just approximations. Unless a number is the square of a whole number to begin with, the square root can only be “more or less” good.

5. Just as finding the square of a number — using the calculator key identified as x2 — is a very important idea in mathematics, so is finding the square root of a number. While the ideas are related, they are very different; so be careful how you use them.

The square root of 16 is 4; but the square of 16 is 256.


Background: II

6. Earlier in your math study you learned how to make factor charts. Here are some examples to refresh your memory.

7. A close examination of these charts will show some patterns.

a) The factors on the left side go in small to large order, while on the right side it is the reverse.

b) The factor pair at the bottom of each chart is the pair with the smallest difference of any other pair.

c) The last factor on the left side is either the square root of the chart’s number or is less than the square root of the chart’s number.

36 = 6 and100 = 10 for the first and last charts. 45 = 6.7, 72 = 8.4, and 84 = 9.1 (approximately) for the others.

8. Those patterns make things easier for us when we wish to find all the whole number factors for a number. And we will use them when we start to test large numbers to see if they are prime or not.

9. You see, you were asked to memorize the primes from 1 to 100. But to ask you to memorize beyond that level is for sure “a bit much”, don’t you agree? However, with a calculator, the primes from 1 to 100, and paper-&-pen, we can decide if numbers that are much, much larger are prime or not. In the next page, you will learn how to test for primeness with numbers like 1271, 1653, 2009, and so on.

10. One final point: Our search for primes of any size will be made more efficient if we remember that, after 2 and 5, primes can only end with the digits 1, 3, 7, or 9. Without that, the number CANNOT prime. On the other side of the coin, ending with those digits does not mean a number IS a prime!


The Lesson

1. Let’s begin with an example done step-by-step.

Is 2111 a prime or not?

a) First, we note that it might be a prime because it ends with the digit “1”. (This also means that 2 and 5 cannot be factors.)

b) Next, by adding the digits — 2 + 1 + 1 + 1– we get 5. That is not a multiple of 3, so 2111 is also not a multiple of 3.

c) This leaves us with the task of checking other numbers to see if they might be factors. But, don’t worry, it’s not as hard as it sounds. We only have to use primes.

d) How many primes? Just up to the square root of 2111, and not any further.

2111 = 45.9 (approx.)

So we will use only the primes from 7 to 43.

2. We will build a chart like before. Put the primes from 7 to 43 in the left side, get out our calculator and get to work — dividing!

-----|------       You must observe one important thing about the
7    |  301.5   quotients.  Since our main concern is whether or
11   |  191.9   not it is a whole number -- thus proving the number
13   |  162.3   is not prime -- I have decided it is sufficient to
17   |  124.1   merely truncate in those cases in which the quotient
19   |  111.1   comes out "in decimals".  Besides that, it's easier
23   |   91.7   than having to remember any rounding rules.  Okay?
29   |   72.7
31   |   68.0      In this example, we see that 2111 is a prime,
37   |   57.0   because no quotient was a whole number.
41   |   51.4
43   |   49.0   That's all there is to it!

3. Let’s try another number: 1003. Since

1003 = 31.6 (approx.), we only need to try the primes up to 31. (And if we’re lucky, we won’t even have to go that far!)

----|-------   	You see?!  This time we found a whole number
7   |  143.2	quotient when we divided by 17, so we didn't
11  |   91.1	have to go all the way up to 31.
13  |   77.1
17  |   59	So 1003 is NOT prime; it is 17 × 59.

Teacher Tips

This ends the three handout sheets. From here on, the lesson may take any form you wish:

  • testing a given set of numbers to see if they are prime or not;
  • ask students to find 5 or 6 primes in this range of their own choice, providing the charts to support the results;
  • make a contest to see who can find the most primes, say from 1000 to 2000, for homework.

When I make a little quiz on this, I use about five numbers, two of which are prime and the rest are not — just to keep things honest. In the final analysis, I feel this work makes a “good problem” that takes more than just a minute or so to do, involves important number theoretic ideas, and is made feasible by the intelligent use of the calculator.

In Part II, mention was made about memorizing the primes from 1 to 100. This is not as great a task as it may sound at first. There are only 25 of them, ten of which are less than 30. And when one knows the divisibility tests for 2, 3, 5, and 7, the memorization is virtually automatic. To ease the task, the 25 primes can be grouped like this:

2, 3, 5, 7 53, 59
11, 13, 17, 19 61, 67
23, 29 71, 73, 79
31, 37 83, 89
41, 43, 47 97

Update (Feb. 2002)

Researchers Discover Largest Multi-Million-Digit Prime

Using Entropia Distributed Computing Grid.

213,466,917 – 1 is now the Largest Known Prime.

For more information, click http://www.mersenne.org/13466917.htm.

Sexy Primes


Before you begin to think that this page of WTM is going to be “x-rated”, we urge you to put that thought out of your mind right
now. You see, we have merely created a new category for classifying numbers. (We assure you that we’re not going to talk about boy-numbers and girl-numbers.) Rather we will do this with that very important kind of number, the prime number.

But first a little review of well-known topic in basic number theory.


As all good math students know, a pair of twin primes is simply two prime numbers that have a positive difference of 2. For
those who have forgotten this fact, a couple of examples should suffice to jog their memories: {11, 13} and {29, 31}.

There are many, many pairs of twin primes out there in that big ocean of numbers. Why not go fishing for some of them…

Now something not so well known will be demonstrated here by way of introducing a new definition for categorizing prime numbers.

First, we begin by writing out some of the natural numbers in rows of six numbers.

1 2 3 4 5 6
7 8 9 10 11 12
13 14 15 16 17 18
19 20 21 22 23 24
25 26 27 28 29 30
31 32 33 34 35 36
37 38 39 40 41 42
43 44 45 46 47 48

If one observes this array of numbers carefully, it will be clear that prime numbers only appear in the first and fifth columns — that is, after we get past the two primes of 2 and 3. In fact, if
you consider the math involved, it should be obvious. The second, fourth, and sixth columns are composed of only even numbers; so that takes care of them in short order. (Why can we eliminate the 3rd column almost as easily?)

However, just because a number is in the 1st or 5th columns doesn’t mean that it is prime. All we’re saying is that if a number is a prime, it can only be found in one of those two columns.


Now we’re ready to explain what is “sexy” about all this. Notice that sometimes in two consecutive rows two primes appear “one
above or below another.” Find 13 and 19 to see what we mean. And the positive difference of the two primes is (you’ve guessed it!) 6. If you recall your Latin number words — no, we didn’t say Roman numerals –, “sex” is Latin for “six.” That’s why we have the name of sextillion in our work with large numbers; and sextuplets for the birth of 6 babies at once.

So, there you have it. If a pair of primes has a positive difference of 6, we here at WTM have declared that such primes shall
henceforth and forevermore be called SEXY PRIMES. How many sexy primes can you find?

Of course we should not limit ourselves to just two primes at a time. We can have sets of 3, 4 or even 5 primes that are sexy.
These groups come to mind: {31, 37, 43}, {251, 257, 263, 269}, and {5, 11, 17, 23, 29}.

* *** *

After writing the above information and publishing it in my November 1997 issue of “Trotter Math” News, I received the following e-mail message from Monte Zerger, a math professor from Adams State
College in Colorado:

I find your “sexy numbers” fascinating and felt compelled to investigate them a bit. Here are some things you may have already

1. It is impossible to have more than four consecutive sexy primes, except for the 5, 11, 17, 23, 29 you mention. This is because the unit’s digit of numbers of the form 6n – 1 or 6n + 1 will cycle through the odd digits, that cycle being either 7, 3, 9, 5, 1, 7, 3, … in the case of 6n + 1 numbers or 5, 1, 7, 3, 9, 5, 1, … in the case of 6n – 1 numbers. Thus every fifth number of the form 6n – 1 or 6n + 1 is divisible by 5.

2. From this it is easy to see that a string of four consecutive sexy primes must begin with a prime whose unit’s digit is 1.

3. For lack of a better idea at the moment, let’s call such a string of four sexy primes a “sexy foursome.”

4. Looking at years, the last sexy foursome was (1741, 1747, 1753, 1759) and the next one won’t be until (3301, 3307, 3313, 3319).

5. However there’s something rather sexy about this century. It began with a sexy threesome (1901, 1907, 1913) and ends with another (1987, 1993, 1999). There were no other sexy threesomes in this century.


Let’s return for a moment to the matter of twin primes. What is the strongest statement that we can say about the set of numbers that lie between twin primes? Oh, it’s easy to say that they are all
even numbers, but that’s not very exciting, is it? Let’s go for more!

There is one set of 3 primes that might be called “triplets”, because they have two differences of 2 among them. We’re thinking of {3, 5, 7}, of course. Can you prove that these three primes are the
only ones to have this property?

Finally, as we saw above, there was a set of sexy primes made up of 5 primes. Can you prove that there could never be a set of 6
primes, coming from 6 consecutive rows and in the same column?