# 13

This number is a prime.

Photo credit: Patrick Kik |

2^{13} - 1 is a Mersenne prime.

There are 13 Archimedean solids.

The smallest emirp.

13^{2} = 169 and its reversal 31^{2} = 961.

13 is the only prime that can divide two successive integers of the form *n*^{2} + 3. [Monzingo]

13^{13} - 13 + 1 is prime. [Luhn]

A "baker's dozen" is a group of 13. Its origin can be traced to a former custom of bakers to add an extra roll as a safeguard against the possibility of twelve weighing light.

There are 13 tricks in each hand of bridge. [Goren]

13^{2} divides 12! + 1.

The smallest inconsummate number in base 2. [Conway]

13 is half of -1^{3} + 3^{3}. [Stark]

The olive branch on the back of a U.S. one-dollar bill has 13 leaves. [Weirich]

The number (coincidentally prime) of known positive integers which are not the sum of two semiprimes. [Capelle]

The smallest 2-digit number that produces the most primes by altering one digit of decimal expansion of n (without changing the number of digits). A set of eight primes can be produced in this manner {11, 17, 19, 23, 43, 53, 73, 83}. The sequence of such n-digit numbers starts 1, 13, 357, 4221, 42189, 344253, 1851759, 15848679, ... . [a(4)-a(8) by Metin Sariyar of Turkey]. See largest known. [Honaker]

The United States flag once had 13 stars and 13 stripes, which represented the 13 original colonies.

The smallest prime whose digits are reversed in base (1 + 3). [Haga]

There is no elliptic curve over the rationals *Q* having a rational point of order 13. [Mazur and
Tate]

13^{2} = 7 + 8 + ... + 19. [Bertuello]

13^{2} can be expressed as the sum of 1, 2, 3, 4, 5, 6 distinct squares. [Crespi de Valldaura]

The longest recorded flight of a chicken is 13 seconds.

131211109876543212345678910111213 is prime. [Kulsha]

13^{2} turned upside down is prime.

ELEVEN + TWO = TWELVE + ONE. Can you solve this doubly-true anagrammatical equation?

13 is the smallest prime which is the sum of two distinct prime squares (2^{2} + 3^{2}). [Gevisier]

Euclid (circa 300 B.C.) and Diophantus (circa 300 A.D.) each composed 13 books.

In May 2001, Ford announced that it would replace 13 million tires on its Ford Explorer SUVs. [Caldwell]

Using the first three primes we have: 2^{3} + 5 = 13. [Trotter]

Alfred Hitchcock's directorial debut was the film *Number 13*, which was never completed. [Liebert]

The term paraskevidekatriaphobia was coined by therapist Dr. Donald Dossey from the Stress Management Center/Phobia Institute in Asheville, North Carolina, and refers to the fear of Friday the 13th. He claims that when you learn how to pronounce the word (*pair-uh-skee-vee-dek-uh-tree-uh-FOH-bee-uh*), you'll be cured of the affliction.

Osama bin Laden is written with 13 letters. [Gevisier]

A Turing Machine with four conditions can write a maximum of 13 characters. [Luhn]

The Beatles recorded 13 albums during the time in which they were together as a group. [Litman]

The sum of primes up to 13 is equal to the 13th prime. [Gupta]

*M*(13) can be expressed as the sum of 13 consecutive primes (599 + 601 + ... + 661). [Vrba]

The sum of the remainders when 13 is divided by all the primes up to 13. [Murthy]

The smallest prime whose sum of digits is a square. [Gupta]

The Munster's, in the 1960's America television comedy, lived at 1313 Mockingbird Lane.

Euler had 13 children, only 5 of whom reached adolescence and 3 of whom survived him. [Poo Sung]

prime(13) = reversal(sigma(13)). It is the only multidigit number with this property. [Firoozbakht]

The sum of the fourth powers of the first 13 primes plus one is prime (6870733). [Patterson]

The 13th of May 2011 will be a "double Friday the 13th," i.e., the sum of the digits of 5/13/2011 equals 13. The next time this will happen in a prime year is 1/13/2141. [Caldwell]

The Jewish sage Moses Maimonides established 13 principles of the Jewish faith during the Middle Ages. [Croll]

Ironically, 13 is a lucky number. [Emmert]

The largest known Fibonacci prime index of a Fibonacci prime. [Rupinski]

A concatenation of the first two triangular numbers. [Gupta]

The sum of first 13 prime numbers is 238 (whose sum of digits is 13). [Gupta]

The smallest prime that can be expressed as the sum of two primes (2 + 11) and two composites (4 + 9) in only one way. [Gupta]

There are exactly 13 regular rank four locally projective abstract polytopes. [Hartley]

The largest minimal prime in base 3. [Rupinski]

13 is the smallest prime that cannot be a member of a Sophie Germain pair. [King]

13333333333333/13 is prime. The numerator has thirteen 3's following 1. [Poo Sung]

2^{13} - 13 is prime. [Schlesinger]

Beethoven's Sonata in C min "Pathetique" is Opus 13. [Necula]

The unique prime sum of squares of two consecutive primes (2^2 + 3^2). [Gallardo]

The dice game Yahtzee consists of 13 rounds. [Bailey]

Three planes can cut a donut into a maximum of 13 parts. [Laurv]

13^{2} = 8^{3} - 7^{3}, i.e., one solution of the diophantine equation z^{2} = x^{3} - y^{3}. [Jun Mizuki]

The U.S. House has voted a total of 13 articles of impeachment against U.S. Presidents: 11 against Andrew Johnson and 2 against Bill Clinton. 13, 11, and 2 are all prime. [Wolfe]

A female Virginia opossum usually has 13 nipples: twelve efficiently arranged in an open circle with one in the center. The length of gestation for this curious marsupial is about 13 days. [McCarthy]

Apollo Lunar Mission number 13, which was aborted while enroute to the Moon because of an explosion of a fuel cell in the service module, left the launch pad at 13:13 (CST) hours military time and the accident occurred on April 13. [Cooley]

The first multidigit half-period prime, i.e., those for which the period equals (*p* - 1)/2.

The 13th Mersenne prime is 2^{521} - 1 and the 13th Lucas number is 521. [Poo Sung]

π(13) = 1! * 3!. [Gupta]

The fear of Friday 13th can be traced back to the suppression of the Templars and the confiscation of their property by Philip IV of France on Friday 13th in October 1307. [Bassano]

The number of 13-digit primes is an emirp. Recall the smallest emirp. [Green]

The first Fibonacci emirp. [Silva]

Patau syndrome, also known as trisomy 13, is the consequence of a rare chromosomal abnormality. [Smith]

Girolamo Cardano (1501-1576) divided cubic equations into 13 types (excluding x^3 = c and equations reducible to quadratics) in his work *Ars Magna*, which was the first Latin treatise devoted solely to algebra. [Poo Sung]

Write thirteen 13's and delete the last digit. Luckily, you get a prime. [Avrutin]

The only known multidigit number *n* such that sigma(*n*) is the reversal of prime(*n*). [Capelle]

In five-card poker, the chance of holding two pairs or a better hand is approximately 1/13. [Post]

13 is the smallest prime having only one 1. [Poo Sung]

13 is the hypotenuse of the unique Pythagorean triangle whose area numerically equals its perimeter. [Beedassy]

13 is the smallest prime number p such that (p+2)^2 divides the least common multiple of the squares of the p first numbers. [Rodrigo]

There are 13 composite numbers between 13 and its reversal. [Silva]

The longest name of a book of the KJV Bible is Thessalonians with 13 letters. [Silva]

A three-digit number *abc* is divisible by 13 if 13 divides
*a* + 4*b* + 3*c*.

The only prime subscript of a triangular number that is
also a square pyramidal number : 1 + 2 + 3 + ... + 11 + 12
+ 13 = 1^{2} + 2^{2} + 3^{2} +
4^{2} + 5^{2} + 6^{2}. [Beedassy]

The only Wilson emirp. [Loungrides]

The smallest prime formed from consecutive odd digits. [Silva]

In the Walt Disney movie "The Computer Wore Tennis Shoes," Dexter Reilly is asked to raise 13 to the 9th power (and then take the square root of this).

The number of brilliant Harshad (or Niven) numbers in base 10. [Capelle]

The smallest idoneal emirp. [Loungrides]

The smallest emirp Delannoy number. [Loungrides]

In 2007, 13-digit International Standard Book Number (ISBN) were introduced. [McCranie]

The floor function of e^e - phi^phi, where phi is Golden Ratio. [Firoozbakht]

5*11 - 2*3*7 = 13 (first six primes). [Firoozbakht]

13!+31!-1 is prime. [Silva]

The prime whose square is equal to the sum of squares of
all digits in which primes end: 13^{2} =
1^{2} + 2^{2} + 3^{2} +
5^{2} + 7^{2} + 9^{2}. [Beedassy]

Primelephas was an extinct genus of elephants and mammoths that averaged 13 feet in length.

At most 13 baseball players can be in play: 9 fielders, 3 runners, and 1 batter. [Poo Sung]

The largest prime *p* such that *p* modulo
*q* is square-free for any prime *q* below
*p*. [Beedassy]

There are 13 positive numbers n such that n is equal to the number of 2's in the decimal digits of all natural numbers smaller or equal to n. [Capelle]

Counting 0 and 1 as the first two members of the Fibonacci series, term 13 is a perfect square (144). [Fellows]

PG-13 is a movie rating that admits all ages but suggests parental guidance of children under age 13. [Teague]

The lesser member of the smallest twirp, i.e., a emirp pair such that each member also belongs to a twin prime pair. In this case, the associated primes are 11 and 29. [Green]

The smallest prime sum of semiprimes. [Silva]

The only Fibonacci prime equal to the sum of the squares of two consecutive Fibonacci primes. [Silva]

13 = (1*3)^0 + (1*3)^1 + (1*3)^2. [Asa]

The smallest prime number having exactly one representation as sum of squares greater than one. Note that its reversal is the largest one. [Capelle]

There are exactly 13 positive integers that cannot be written as the sum of four hexagonal numbers. Their sum is a prime number whose sum of digits is 13. [Capelle]

13^13 - π(13)^π(13) is prime. It is the largest known prime with this property. The next prime of this form (if it exists) will have more than 50000 digits. [Firoozbakht]

The smallest prime such that if its digits are rendered in a prime base p, the result, 13 (base p), is never prime. [Green]

There are 13 cards of each suit in a deck of playing cards. [Green]

The smallest prime that when concatenated to its Collatz
sequence (__ 13__ --> 40 --> 20 -->
10 --> 5 --> 16 --> 8 -->

__) inexorable ending 4-2-1 forms another prime: 13421. Note that a reverse concatenation also yields a prime (12413). [Beedassy]__

**4-2-1**Metatron's Cube is created by joining the centers of 13 circles arranged in a "Flower of Life" pattern.

If we count all the traditional constellations, there are 13 in the zodiac. [Graham]

"COSMOS" by Carl Sagan was originally a 13-part television series.

The smallest prime that is equidistant by a prime (3)
between the sum of squares of its digits (1^{2} +
3^{2}) and the square of the sum of its digits (1 +
3)^{2}. [Beedassy]

'Friday the 13th' is a franchise that consists of several movies, a television show, novels, comic books, and related merchandise. Submitted on Friday the 13th. [Green]

The Dynasty of Primes was formed by the first 13 Cybertronians. [Jetfire]

The smallest prime whose index is a perfect number. [Pol]

The product of 13 and seven is a perfect number in Abbott and Costello's math spoof (from "In The Navy," 1941).

13 can be partitioned in 101 ways. Note that π(101) + π(13) - 1 equals the reversal of 13. [Jevtic]

13 squared equals the sum of the first five emirps. [Silva]

Oprah Winfrey ran away from home at the age of 13 and was sent to a juvenile detention home, only to be denied admission because all the beds were filled.

The number of five-digit primes formed from the five nonzero nonprime digits. [Silva]

13+10(1+2+3+4+5+6+7+8+9+10+11) is prime for each and every summation from 1 to its twin prime pair 11. [Haddish]

In the great ancient Indian epic *Mahabharata*, the
righteous Pandava clan, after losing their staked kingdom
through deceit in a game of dice schemed out by the envious
Kaurauva cousins, were forced to undergo 13 years of exile. [Beedassy]

Mice have only 13 pairs of ribs. [Post]

Religious scholar A. C. Bhaktivedanta Swami Prabhupada gives the equivalent length of a yojana as approximately 13 kilometers. [Beedassy]

"Remarkably, it turns out that for N = 13 the set of operators of 'exceptional' type (4) is empty, i.e., any Hamiltonian operator of order N = 13 can be taken by a contact transformation to an operator of type (1), (2) or (3)! We conjecture that the same holds for all N > 13." Reference: On Classification of Poisson Vertex Algebras by Alberto De Sole, Victor Kac, and Minoru Wakimoto. [Post]

π(π(1) + π(2) + ... + π(13)) = sigma(13). It is the only number with this property. [Firoozbakht]

A coven usually consists of 13 witches. [Cheek]

Can you find a total of 13 triangles in this figure? [Thorn]

13 is the smallest prime p such that p^2 and (p+1)^2 are anagrams (next such prime is 157). [Seidov]

13 describes its neigboring prime gaps, before and after. There is 1 composite before it and 3 composites after it. [Honaker]

ELEVEN PLUS TWO and TWELVE PLUS ONE both contain 13 letters. [Homewood]

In chess, a Bishop controls a maximum of 13 squares of a given color. [Beedassy]

There are 13 basic points of belief to which Mormons subscribe.

There are exactly 13 convex tangrams . [Gudipati]

In 2013 there were two 'Friday the 13ths' exactly 13 weeks apart. [Homewood]

There are precisely 13 nonnegative cube integers, ranging from 0 through 970299, for which the final digit or the final two digits are the cube root. [Post]

The number of atoms in Cytosine. [Pol]

The only prime less than a googol of form (2^x+1)/5, (case x=6). [Loungrides]

Currently (between the years 1901 and 2099) the Julian calendar is 13 days behind the Gregorian calendar. [Gupta]

'Mercury 13' refers to thirteen American women who underwent some of the same physiological screening tests as the astronauts selected by NASA for Project Mercury. [Yonce]

The square of the prime 13 is 169 and 169 is the concatenation of the squares of two consecutive numbers 4 and 3. This is the largest prime known with this property. [Rivera]

The number of prime years on "Passing Time" by sculptor Anton Parsons. [Spragg]

13 + (1+3) is the next emirp. The only two-digit emirp with that property. [Silva]

There are 13 cubes that are each the concatentation of 2 five-digit prime numbers. [Gaydos]

This is the only prime p such that 2*p+1 is a cube. [Gupta]

A path on a chessboard from c5 to d4 in 13 straight queen moves that cover all squares is impossible.

Possibly the only positive integer that is prime, lucky, happy, and Fibonacci.

A February 'Friday the 13th' in a leap year occurs 13 times within a 400-year period. [Gupta]

The only known prime of the form 12^n+1 with integer n >= 1. [Luen]

Pick a number n ≡ 2,4,7 (mod 9). Square n, then add the digits of the result; repeat. Eventually, you will reach 13. E.g., 2^2=04. 0+4=4. 4^2=16. 1+6=7. 7^2=49. 4+9=13. Also, 2+4+7=13. [Missailidis]

ROT13 is a popular encryption method on the Internet since it obfuscates text sufficiently to prevent accidental reading of the text. [Macura]

sod(M14 exponent) = sod(M17 exponent) = sod(M18 exponent) = sod(M20 exponent) = 13. [Harrison]

13 is the smallest prime p such that p+(p-1)*(p-2) is not a prime. [Bui Quang Tuan]

The number of primes formed by inserting two zeros between the digits of a double-digit number: 1009, 2003, 3001, 4001, 4003, 4007, 5003, 5009, 6007, 7001, 8009, 9001, 9007. [Loungrides]

There are 13 three-digit squares without repeated digits.

The smallest prime equal to a prime plus its digit sum, here 11 + (1 + 1). [Hasler]

The smallest expansive prime, i.e., a prime from which another can be obtained by repeating its digits in order, each digit repeated d times. [Silva]

Arsenal F.C have won the English F.A. Cup a record 13 times. They have also won 13 League titles. [Homewood]

A four-digit number abcd is divisible by 13 if 13 divides a + 4b + 3c + 12d. d is calculated using the coefficients of a + 4b + 3c which shows if a three-digit number is divisible by 13. b * c / a = 12 and a five-digit number is divisible by 13 if 13 divides a + 4b + 3c + 12d + 9e because the coefficients of c * d / b = 9. multiplying the last two coefficients and dividing by the one before them will always give the next term in this equation and therefore any number of any length can be checked for divisibility by 13. [Homewood]

All digits 1 thru 9 appear in the first 13 decimal digits of π. [Homewood]

The smallest Republican prime. [Hilliard]

Floor function of π*e*phi, where phi is the golden ratio. [Gupta]

There are thirteen 4-digit primes each consisting of four successive digits, i.e., 1423, 2143, 2341, 2543, 4231, 4253, 4523, 4567, 4657, 5647, 5867, 6547, 6857. [Loungrides]

After needing 13 liters of blood for a major surgery during his teens, Australian James Christopher Harrison pledged to return the favour eventually becoming world record blood donor after the discovery in his blood of a unique antibody that cures Rhesus disease thereby saving millions of babies over 6 decades. [Beedassy]

The 13 end-digits of the overwhelmingly gigantic Graham's
number (*g*_{64}) form a second prime
concatenation: 7262464195387. [Beedassy]

The *Art Class Curator* provides 13 ways to integrate art & math.

13 is the sum of the third prime and third composite number. [Hollesen]

The longest known sequence of consecutive primes with distinct digits starts from 13. [Gupta]

The *only known* cuboctahedral prime. [Wiszowaty]

The first member of the smallest pair of cousin emirps. [Silva]

13 is the smallest prime that remains prime when its digits d are changed to d^n, for n = 2, 3 and 4, one at a time: for n=2, 13->19; for n=3, 13->127; for n=4, 13->181. [Rivera]

There are 13 squares with distinct digits that have the same sums of digits as their square roots. [Gaydos]

Divide 13 by all the primes less than 13 and add the sum of the remainders: 1+1+3+6+2 = 13. This never happens again. [Bergot]

The first sum of two squares where the two numbers are prime and the sum is also prime. (2² + 3² = 13). [Homewood]

The Tahitian alphabet contains only 13 letters. [Azuz]

Sven Freitag developed a wonderful free app called "Prime Cracker" for quickly finding out whether or not a given number is prime. Requires iOS 13.0 or later on an iPhone.

The Gregorian Calendar is named after Pope Gregory XIII. [Harrison]

Taylor Swift's favorite prime number is 13. "Swifties" have
their own conspiracy theory:

* The date of Super Bowl LVIII in 2024 is 2/11; 2 + 11 = 13.

* This is Super Bowl 58; 5 + 8 = 13.

* The Chiefs are playing the 49ers and 4 + 9 = 13.

* If she attends this Super Bowl, it will her 13th time attending a Kansas
City Chiefs game.

* If she flies to Vegas from Tokyo, the flight will be
roughly 13 hours.

* Taylor was born on December 13.

* She turned 13 on a Friday the 13th.

* Her first No. 1 song had a 13-second intro.

* Her first album went gold in 13 weeks.

There are thirteen prime outputs among the first one hundred terms in the Tribonacci sequence TRI (n) which is recursively defined as follows: TRI (1) = TRI (2) = TRI (3) = 1 and TRI (n) = TRI (n-3) + TRI (n-2) + TRI (n-1) for n>=4. The thirteen prime outputs 3, 5, 17, 31, 193, 653, 1201, 46499, 532159, 3311233, 128199521, 20512526282340991, and 6036724301884645488192191 corresponding to the fourth, fifth, seventh, eighth, eleventh, thirteenth, fourteenth, twentieth, twenty-fourth, twenty-seventh, thirty-third, sixty-fourth, and ninety-sixth terms respectively. [Schiffman]