# 31

This number is a prime.

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π^{3} is close to 31. [Luhn]

The Mir Space Station docked with 31 spacecraft during its history.

31 is the third Mersenne prime.

The number of letters (in English) required to write the word names of the first six primes is the sixth prime reversed, i.e., 31. [Trotter]

There are only 31 numbers that cannot be expressed as the sum of distinct squares.

Portal 31 is Kentucky's first exhibition coal mine.

31 + the floor function of π * 10^{999} is prime. [La Barbera]

The minimal number of moves required in a Tower of Hanoi puzzle containing five discs. The general solution to this puzzle requires a Mersenne number (i.e., 2^n - 1) of moves, where *n* is the number of discs.

The smallest multidigit prime whose reciprocal has an odd period for its decimal expansion.

Mersenne prime *M*(31) could easily be confused with our nearest large neighbor galaxy Andromeda, i.e., M31 (or Messier 31).

31 = 2^{2} + 3^{3}. [Kulsha]

The smallest prime whose digits are reversed in base 28. [Haga]

The smallest prime such that replacing each digit *d* with *d* copies
of the digit *d* produces a different prime (3331).

π(31) is prime. [Millard]

There are 31 milligrams of cholesterol in a tablespoon of butter.

The only known Mersenne emirp.

The smallest prime which is the sum of two Smith numbers (4 + 27).

31 = -1 + 2^{3} * 4, demonstrating four basic arithmetic operations. [Kulsha]

The speed limit in downtown Trenton, a city in western Tennessee, is 31 miles per hour.

31 divides 31^{1} + 31^{2} + 31^{3} + ... + 31^{30} + 31^{31} a prime number of times. [Luhn]

31 letters of the Russian alphabet are pronounceable.

3 + 5 + 7 + 11 + ... + 89 = 31^{2}, and the sum of the first 31 odd primes is a prime square.

Paul Dirac predicted antimatter and won a Nobel Prize in 1933 (a prime). Carl Anderson detected antimatter and won a Nobel Prize three years later. Both men were age 31 when they won the Nobel Prize, and they tie for being the youngest Nobel laureates. [McCranie]

There are 31 pairs of spinal nerves in the human body. [Beedassy]

In the Cambridge segment of Crown & Country TV series, KQED, hosted by Edward Windsor, it was noted that Cambridge University eventually consisted of 31 colleges. [Haga]

The sum of digits of the 31st Fibonacci number is 31. [Gupta]

31 is the largest primorial prime less than the next prime squared, i.e., 5# + 1 = (2*3*5) + 1 = 31 < 7^{2} = 49. [Nicholson]

California was the 31st state to be granted statehood. Hence, the U. S. flag was changed to display 31 stars and 13 stripes (an emirp pair). [Green]

π(π(π(π(31))))=2 is prime, π(π(π(31)))=3 is prime and π(π(31))=5 is prime. [Olsen]

3 + 1 = 4 and (4!/.4 + sqrt(4))/sqrt(4) = 31. [Hoge]

The big "31" sign made its debut at all Baskin-Robbins stores in 1953, offering customers a different ice cream for every day of the month. Note that 31 is the largest prime factor of 1953. [Coneglan]

31 is a prime number equal to the sum of the decimal digits of its 7^{th} power (31^{7} = 27512614111, which sums to 31). [Keith]

The first chapter of the Bible has 31 verses. [Silva]

The first U.S. space satellite (Explorer-I) weighed just less than 31 pounds and was launched on Jan 31, 1958. The high-power transmitter worked for 31 days. [McCranie]

(2^31-1)/(2-1) and (2^31+1)/(2+1) are both prime numbers. [Dobb]

The prime number 31 is also a repunit in three different bases: 31 is 11111 in base 2; it is 111 in base 5, and it is 11 in base 30. [Dobb]

31 and the 31st prime are both Mersenne primes. [Wu]

The smallest multidigit quintan prime: p = (x^5 - y^5 )/(x - y). [Russo]

Buddha preached on the 31 levels of existence in our universe. [Jen]

31 is the smallest multidigit prime in the decimal expansion of cube root of 31. [Gupta]

If t(n) represents the nth triangular number, then 31 is the smallest prime of the form t(p) + t(q), where p and q are themselves prime. [Rupinski]

The smallest prime that is a generalized repunit in three different bases. [Rupinski]

The largest integer such that the average number of triangular numbers required to express the integers less than or equal to it is exactly two. [Rupinski]

The reverse concatenation of first two triangular numbers is prime. [Gupta]

31 squared when turned upside down is a perfect square. [Friend]

The smallest prime that can be represented as the sum of two triangular numbers in two different ways (21 + 10 and 28 + 3). [Gupta]

For any 200560490130 consecutive whole numbers on the number line, exactly 1021870080 will have the number 31 for a lowest prime factor. [Schuler]

7 + 11 + 13 = 31. [La Haye]

Jackson Pollock's *One: Number 31, 1950* occupies an entire wall by itself at the Museum of Modern Art, New York City. The painting is oil and enamel on unprimed canvas. (Studies have shown that some of Pollock's works display the properties of fractals.)

The earliest and the only known case such that the sum of the divisors of two distinct numbers (16 and 25) is the same prime quantity (31), that is to say: 1+2+4+8+16 = 31; 1+5+25 = 31. [Rivera]

(31^{31} - 1)/(31 - 1) is prime. [Capelle]

3^31 - 2^31 is prime. [Capelle]

The smallest prime p such that neither 6p-1 nor 6p+1 are primes. [Necula]

The number of regular polygons with an odd number of sides that can be constructed with compass and straightedge.

(31^31-1)/30 is prime and the largest known generalized repunit prime, where b = n.

The smallest prime that starts a run of exactly four distinct primes that remain prime after each digit d is replaced by d^3: 31, 271, 83431, 5122764271. [Rivera]

Lance Armstrong, in 2003 (at the age of 31), became the 2nd person ever to win the Tour De France 5 times consecutively, beating the runner-up Jan Ullrich by 61 seconds. Note that 2, 5, 31, 61, and 2003 are all prime numbers. [Hartley]

The prime number 31# + 1 has sum of digits equal to 31. [Gallardo]

The sum of digits of all primes less than or equal to 31 is prime. [Patterson]

In the *Star Trek* episode "Terra Prime," Reed's Section 31 contact provides information to help the *Enterprise* crew infiltrate a Martian colony.

After removing two opposing corner squares on a chessboard it is impossible to arrange 31 dominoes in such a way that they cover all the remaining squares.

A computer analysis by Ströhlein and Zagler shows that the winning process in a Queen versus Rook endgame should take at most 31 moves.

Frank Leon Drew was the senior supervising engineer on the Hooker 100" telescope which was the largest telescope in the world for 31 years. [McGown]

Moser's circle problem asks to determine the most pieces into which a circle is divided if *n* points on its circumference are joined by chords with no three internally concurrent. The first few values are 1, 2, 4, 8, and 16. What do you think the next term will be? [Post]

If a rhombic tricontahedron is properly numbered with ascending and descending ordered number pairs, then each sum of the individually opposite paired sides is 31. [McGown]

Ben Johnston is known for composing music based on a flexible tuning system that derives pitches from as high as the 31-limit. The "prime limit" of an interval or chord in just intonation is the largest prime number in its factorization.

Time dilation remained an unproved theory for 31 years after it was first proposed by Albert Einstein. [Wilson]

31 is the largest integer *n* such that the first *n* digits of π after the decimal point are all nonzero. [Keith]

The only two-digit prime that yields an emirp if the next prime is sandwiched between its digits: 3371. [Silva]

The smallest Gaussian emirp. [Loungrides]

31 is the number of the French department of Toulouse where lived Pierre de Fermat, 'Prince of Amateurs' and father of modern number theory. [Beedassy]

The smallest primorial emirp. [Loungrides]

In the movie 2010, the code to destruct HAL is to enter 999, square root, and press integer on a calculator. That gives 31. [McCranie]

31111111111111111111111111111111/31 is prime. The numerator has 31 1's. [Firoozbakht]

31^{2} = sigma(20^{2}). [Wesolowski]

The largest known prime number such that the digits of its square are in descending order. [Capelle]

Steve Homewood from England discovered that the formula 2n^2 + 4n + 31 generates prime numbers over 43 percent of the time for all positive values up to 10 million, thus solving the Alpha31 problem posed by Vasilios Gardiakos.

The largest known prime number p such that the decimal
expansion of 3^{p} does not
contain any zero. [Capelle]

The smallest power of 2 with a pandigital factorization: 536870912 * 4 = 2^31. [Post]

The number of prime facts investigated by Dick Sietz in
his article "Prime List of Primes" that appeared in the *Virginia Mathematics Teacher* (Vol. 30, No. 2, 2004).

On Landon Curt Noll's web page, "The English Name of a Number," he states that: With the following 31 words, we (are) able to count up to 999999999: zero one two three four five six seven eight nine ten eleven twelve thirteen fourteen fifteen sixteen seventeen eighteen nineteen twenty thirty forty fifty sixty seventy eighty ninety hundred thousand million. [Green]

The smallest of three consecutive primes, (31, 37, 41), such that the product of the first two plus 1 is divided by the 3rd. Note that it is the largest known such case. [Loungrides]

31 = (2^{5}-1)/(2-1) = (5^{3}-1)/(5-1) = (6^{3}+1)/(6+1). [Ordowski]

31= 3! + 1! + (3+1)! [Kartiks]

There are exactly thirty-one 8-digit strobogrammatic primes: 10069001, 10911601, 11069011, 11096011, 11869811, 11896811, 16069091, 16188191, 16811891, 16900691, 16969691, 16996691, 18611981, 18688981, 19188161, 19811861, 60088009, 60611909, 60800809, 61096019, 61669919, 61696919, 61969619, 66169199, 66896899, 66911699, 68669989, 68811889, 68969689, 69800869, 69911669.

First occurrence of a run of exactly 31 consecutive integers with an even number of prime factors has never been found.

Just two 1st-move pairs in a chess game yield a maximum of 31 legal 2nd-move options for White: White's double advance of King's Pawn followed by either of 2 double advances resulting in an option to capture. [Merickel]

The sum of the reciprocals of the digits of 2^31 equals 22/7 (grade school approximation of π). [Gaydos]

California is the 31st U.S. state and its postal code is CA, the 3rd and 1st letters of the alphabet. [Gaydos]

There are at least 31 Fibonacci primes. [Homewood]

The only prime of the form n^5-1. [Moehring]

Q: Why is it that programmers always confuse Halloween with Christmas? A: Because 31 OCT = 25 DEC. [Krypted]

31-almost primes recently made an appearance in work related to the twin prime conjecture and questions about the prevalence of other tuples of prime numbers.

The sum of 31 primes starting with 31 of every 31th prime is prime. [Wiszowaty]

There are 31 prime quintuplets of the form {p, p + 4, p + 6, p + 10, p + 12} less than one million. [Homewood]

31 is the first emirp to appear in the primeth sequence:
P_{1} = 2 ; P_{2} = 3 ;
P_{3} = 5 ; P_{5} = 11 ;
P_{11} = 31. The next emirp 9737333 occurs
after 6 further steps and the third one is not below
10^{25}. [Beedassy]

The only occurrence in the sequence of primes of a pair of
consecutive emirps the next one being "generated" from its
predecessor when expressed in octal scale:
31_{10} = 37_{8}. [Beedassy]

The smallest multidigit self prime. [Gupta]

There are 31 known Wagstaff primes, as of 2018. [Homewood]

The 31st prime is preceded by reversal(31) consecutive composite numbers. [Gupta]

The 8-puzzle always can be solved in no more than 31 single-tile moves.

The world famous (and of utmost rarity) first edition and first printing of Charles Baudelaire's "Les Fleurs du Mal" contains a typing error on page 31: "Les Feurs du Mal" instead of "Les Fleurs du Mal." [Olry]

A repunit of 31 ones is a prime number in base 31. [Gaydos]

The 13353rd century has 31 abundant years. The next such century is the 27339th. Are there any centuries with 32 or more? [Gaydos]

31#/2 +/- 2 are the fourth and largest known (as of 2020) "cousin primes" of the type p#/2 +/- 2. (100280245063 and 100280245067). The other three pairs occur for p = 5, 7 and 13. [Rivera]

There are 31 prime numbers that have sums of reciprocals of digits that are less than 1/2. [Gaydos]

**A computer simulation of cosmic voids that pervade our universe**?

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