13 (thirteen) is the natural number after 12 and before 14. It is the smallest integer with eight letters in its spelled out name in English. It is also the age at which children become teenagers.

In mathematics

Thirteen is the sixth prime number; the next is seventeen. 13 is the second Wilson prime. 13 is the fifth Mersenne prime exponent, yielding 8191.

13 is the second star number and the seventh Fibonacci number. As it is an odd-indexed Fibonacci number, it is a Markov number, appearing in solutions to the "Markov Diophantine equation": (1, 5, 13), (1, 13, 34), (5, 13, 194), ...

13 is also the second happy prime, following 7, and the rethorical 1.

Thirteen is the first prime number to be the aliquot sum of two numbers; the cube 27, and the discrete biprime 35 and it is the base of the 13-aliquot tree.

There are 13 Archimedean solids.

13 goes into 999,999 exactly 76,923 times, so vulgar fractions with 13 in the denominator have six digit repeating sequences in their decimal expansions. It is thus the smallest half period prime OEIS: id=A097443

13 is the largest number whose factorial is less than 10¹⁰. This means 13 is often the largest factorial a pocket calculator can represent without scientific notation. (This is not really a special pure mathematical property, because it requires 10 to be the "main base number" self-evidently.)

At 13, the Mertens function sets a new low of -3, subseded later at 31 with a value of -4.

13 is the only positive integer that is the fourth root of the sum of the squares of two successive positive integers (119 and 120).

The 13th root is the most famous integer root calculation record, because 13 is the first prime number over 10 such that the last digit of a 13th integer power is the same as the last digit of its 13th root.

In base 10, the smallest prime with a composite sum of digits is 13.

13 is a repdigit in base 3 (111).

Let $\_p\backslash mathrm\{log\}\_gx$ denote the discrete logarithm according to the prime number $p$, i.e., the number $l$ modulo $p-1$ such that

$g^x\; \backslash equiv\; l\; \backslash pmod\{p\}.$

Then 13 is the only prime number $p$ such that for any two primitive roots $g$ and $h$,

$\_p\backslash mathrm\{log\}\_gh\; \backslash equiv\backslash \; \_p\backslash mathrm\{log\}\_hg\; \backslash pmod\{p-1\}.$

There exists an a periodic set of thirteen Wang tiles.

The Primorial+1 : 13#+1=2*3*5*7*11*13+1 is the first number of the form n#+1 which is composite.

Also, the first whole number to have the sum of its digits equal 13 and also be divisible by 13 is 247 (2+4+7=13)(13*19=247) There are 24 hours in a day and 7 days in a week. The next number to have these attributes is 364 (3+6+4)(13*28) This is one day less than a year. 13 flipped around is 31, which accounts for the number of days in most months.