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.

Prime Curios III: The number 23

Number23.jpg
“The Number 23”, a 2007 movie which starred Jim Carrey and was directed by Joel Schumacher.

23 was the number featured in two movies. The 1999 movie called “23” was in German,  starring August Diehl, Fabian Busch, Dieter Landuris, Jan Gregor Kremp. The Hollywood movie called “The Number 23” was released 8 years later, starring Jim Carrey, Virginia Madsen, Logan Lerman, and Danny Huston. Both movies appear to be psychodramas based in some way upon the fear of the number 23, a condition known as Eikositriophobia. The phrase “the 23 enigma” figures in this film, usually uttered by people who find a mystical significance with the number 23.

Yes indeed there are people in the world that fear a specific number of some kind. Each number has its own long and nearly unpronouncable name. The most common fear would probably be Triskaidekaphobia, the fear of the number 13 (oh yeah, another prime number). But let’s stick to the number 23 for this article.

    • It is part of the expression “23 skidoo!”
    • 23 is a number whose digits add to 5, the next prime.
    • 23 is a part of the Birthday Paradox: 23 is the minimum number of people to have in a room, where the probability of two people having the same birthday is just over 50%. Related to this, is the question of how many ways can 23 people be paired? It turns out that this becomes {23 \choose 2} = 253 \approx 365\ln{2}.
    • Twin primes are a pair of prime numbers which are also two consecutive odd numbers. Examples are 3 and 5; 5 and 7; 11 and 13; 17 and 19. The first prime that cannot be “twinned” in this manner is the number 23.
    • 23 is also called a “factorial prime”, because the next whole number, 24, is a factorial: 24 = 4\times 3\times 2\times 1 =4!.
    • Reversing the digits makes it a power of 2: 32 = 2^5=2^{2+3}.
    • e^\pi, rounded down to the next lower integer, is 23, or: \lfloor e^\pi \rfloor =23.
    • The sum of the first 23 primes is 874, a number divisible by 23.
    • 23 is made from digits which are the first two primes, 2 and 3. Wikipedia calls a number like 23 a Samarandache-Wellin prime, where putting together the digits of the first n consecutive primes makes the next Samarandache-Wellin prime. “2” is the first prime, so is the “sum” of only one prime. The second prime has to be the concatentation of the first two primes, so “23” is the next one. 5 divides “235”, so that doesn’t work; but “2357” is prime, the concatenation of the first 4 primes to make the third Samarandache-Wellin prime.
    • 23 is also the number of chromosomes in a human; the atomic mass of sodium; the degrees of tilt of the Earth’s axis; the Bible’s 23rd Psalm; and the number of times Roman emperor Julius Caesar was stabbed in the Theatre of Pompeii.