Prime numbers are integers that have no divisors except themselves and one.  The sequence of primes begins 2, 3, 5, 7, 11, 13, 17, … and there are infinitely many. Below are ten examples of prime numbers that have unusual properties that you will not learn about in school. Tell your friends or your next Saturday-night date about them and they will surely be impressed with your arcane mathematical knowledge.

379009. When you type this prime into a calculator and turn it upside down it spells ˜Google,’ the name of the popular web search engine.

613. This is a very unusual prime and one of my favorites. You can get three different classes of numbers with this prime by simply rearranging its digits. If you move the first digit to the end you get 136, which is a triangular number (they have the formula  n * (n + 1)/2 and (16 * 17)/2 = 136), and if you move the first digit of 136 to the end you get 361, which is a square: 19^2=361.

433.  This prime is close to the title of a composition by avant-garde composer John Cage. His piece 4’33” (referred to as “four thirty-three”) entails playing nothing at all for four minutes and thirty-three seconds.

81457.  Leetspeak is a kind of coded language used among a segment of the Internet population. If we use a portion of the Leet cipher: 8 = b, 1 = l, 4 = a, 5 = s, 7 = t, the prime 81457 spells the word “blast” in Leetspeak. 

77345993. Another calculator prime. When turned upside down it looks as if the word EGGSHELL is spelled.

6089. This is the beginning of a curious sequence of six primes whose digits contain only circles (0, 6, 8, 9). They have the form 609 * 10^n- 1. They can be arranged nicely in a stack like this:

6089
60899
608999
6089999
60899999
608999999

6823. According to the Jewish Encyclopedia, the Tetragrammaton (YHWH) is one of the names of the God of Israel and it occurs 6823 times in the Old Testament.

15251. This isn’t a prime but it has some curious properties related to them. It’s the least palindromic number (it reads the same way forward and backward) such that the sum of primes from its smallest to largest prime factor is also a prime, and the sum of composites from its smallest to largest prime factor is a prime. I.e., 15251 = 101 * 151, and  101 + 103 + 107 + 109 + 113 + 127 + 131 + 137 + 139 + 149 + 151 = 1367, a prime; and the sum of all composites between 101 and 151:  102 + 104 + 105 + 106 + … (there are a lot of them) … + 147 + 148 + 150 = 5059, a prime.

(6 * 10^(6254+10) + 6881691889) * 10^(6254+1) + 9. This is the expression for a probable prime that has 12,520 digits.  It is called a ˜probable prime’ because although it has passed Fermat tests (I won’t explain those here) no one is sure how to certify the number prime because it doesn’t have an easily provable form. It’s also a ˜strobogrammatic’ prime because it will look the same when rotated 180 degrees: The digits 0, 1, and 8 look the same when turned upside down and the digits ˜6′ and ˜9′ are considered vertical reflections of each other.

(4 * 10^(5819+13) + 3141592653589) * 10^(5819+1) + 3. This is another probable prime and it has 11,653 digits. Notice that the first 13 digits of Pi = 3.141592653589793 … occur in its center. Strange.

And there you have ten unusual and wonderful prime numbers. Memorize their properties and tell your family and friends about them!

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