Prime Curios IV: The number 13

12 is a nice number. It is why we order a dozen doughnuts, for example. You can most likely keep all the guests happy, allowing you to divide the doughnuts among 2, 3, 4, 6, or 12 guests (not counting yourself). Thus, 12 has a lot of divisors, making it a highly composite number. It would have been nice to have our money denominated under a base-12 system (called a “dozenal” system), since there are so many ways to evenly divide the cash.

13 is just one more added to 12, but then the number becomes very uncooperative by contrast. You can’t even divide 13 in half, evenly. While 13 doughnuts is a called “a baker’s dozen”, there is no way to divide them among guests unless you have 12 guests; but at least this way you have a doughnut left over for yourself. It is amazing what happens sometimes when you add 1. But the indivisibility of 13 is a property shared by all prime numbers.

13 objects.

13 is the second number in a series called a star number. A star number is the number of objects you can arrange into a six-pointed star. The next 3 numbers in this series are: 37, 73, and 121. The first number was 1. 13 is likely the only positive number that is prime, lucky, happy, unlucky (depending on the culture), and Fibonacci.

An 18th-century American flag.

13 is the number of archimedian solids; the number of tricks in a bridge hand; the number of leaves on the olive branch of an American dollar bill; the number of both stars and stripes on an American flag from the 18th century; it is the greatest number of seconds that a chicken has been recorded flying; it is also the number of volumes in Euclid’s Elements; the number of record albums recorded by The Beatles; the number of digits in an ISBN number; and the number of rounds in Yahtzee.

13 in base-10 is 31 in base-(1+3) (or base-4). 13 is a factor of the 3-digit number abc if 13 is also a factor of a + 4b + 3c. 13 with its digits reversed is 31, which is also prime (called an emirp). ELEVEN PLUS TWO contains 13 letters; so does TWELVE PLUS ONE. Also, the product of pi, Euler’s constant, and the golden ratio, rounded down, or: \lfloor \pi e \Phi \rfloor = 13. Performing arithmetic on the first 5 primes yields 13 if done in the following manner: (5\times 11) - (2\times 3\times 7)=13.

The Republican logo

13 is the smallest Republican prime number. These are numbers whose left-hand digits (not counting the middle) are prime and the right-hand digits are not. 1 is not actually a prime number, nor is it composite. Democratic primes exist whose numeric patterns are the opposite. Thus, the reversal of 13, or 31, is a Democratic prime.

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Belphegor.

The number 1000000000000066600000000000001, which has the number of the Beast (666), flanked on both sides by 13 zeroes and beginning and ending with 1, is prime, and is called a Belphegor prime. This prime is also palindromic (reads the same backwards and forwards). Belphegor is one of the seven princes of hell, presiding over the deadly sin of sloth. The Belphegor prime is also a “naughty” prime, due to the number of naughts, or zeroes present in the number. This number also has 31 digits, which is 13, reversed.  There are 13 prime numbers between 13 and 31.

In 2013, there were two Friday the 13ths, and both were 13 weeks apart. 13 is the number of days the Julian calendar is behind the Gregorian calendar. ROT-13 is a kind of message encryption which “rotates” all letters in the message around the 13th letter of the alphabet with the effect of preventing accidental reading of internet text messages such as spoilers to novels or movies.

I researched the Prime Curios website; Numberphile on YouTube; and the OEIS library of number sequences for this article.

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