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# 5

There are 5 Platonic solids (convex regular polyhedra).

The smallest prime in the first sexy prime pair (5, 11). Note that it is "sexy" since *sex* is the Latin word for six. [Wilson]

The first prime of the form 6*n* - 1.

Provably the only prime that is a member of two pairs of twin primes. [Pallo]

5^{5} = 38 + 39 + 40 + ... + 87. [Dutta]

There are no known Wall-Sun-Sun primes greater than 5.

The 5th Mersenne prime is -1 + 2^{3} * 4^{5}.

5 starts the smallest set of prime quadruplets, i. e.,
(*p*, *p* + 2, *p* + 6, *p* + 8) are all prime. [Murthy]

The smallest Wilson prime.

The mineral apatite is 5 on Mohs' hardness scale.

A cryptarithm is a type of mathematical puzzle in which most or all of the digits in a mathematical expression are substituted by letters or other symbols. In the case of XZY + XYZ = YZX, the value of Z must equal 5.

The 5th nonzero Fibonacci number.

4^{p} - 3 is prime for *p* = 2, 3, 5, 7 and 11 (the first 5 consecutive primes).

In 1769 Lambert noted that the number of the digits of the repetend of a repeating decimal, 1/*a*,
divides *a*-1 for *a* = 3 or a prime greater than 5.

The only prime solution to the "*triangle = square pyramid *" problem, which involves
the equation 3(2*y* + 1)^{2} = 8*x*^{3} + 12*x*^{2} + 4*x* + 3.

Any *Lucas* number L_{5p}, for any prime *p* greater than or equal to 5, has at least two distinct primitive prime factors. [Jarden]

*n*! never ends in 5 zeros. Note that the first 5 terms in the sequence of numbers of zeros that *n*! never ends in are all prime. The set includes 5, 11, 17, 23, and 29.

The only prime that is the sum of "Siamese twins," i.e., 2 and 3, which are the only pair of primes that are conjoined (have no composite between them). [Gevisier]

WCYB5 in Bristol, Virginia, has been rated America's number one television station.

*Subfactorial* 5 is palindromic.

7 * 10^{n} + 1 and 9 * 10^{n} + 7 are prime for *n* = 1 to 5. [Luhn]

5 requires 5 steps to reach 1 in the 3*x* + 1 (or Collatz) Problem. Paul Erdős said about the Collatz problem: "Mathematics is not yet ready for such confusing, troubling, and hard problems." He offered $500 for its solution. (Lagarias, 1985)

5 is close to 6(Phi^{2})/π. [Moody]

A limerick is a light humorous or nonsensical verse of 5 lines that usually has the rhyme scheme *aabba*. [Luhn]

The fewest number of moves for pawn promotion to occur in chess. [Rachlin]

The smallest odd prime equal to the sum of two squares and to the arithmetic mean of two squares. [Capelle]

The word PRIME contains 5 letters. [Russell]

5 is the 5th digit in the decimal expansion of π = 3.1415.... [Gupta]

Every type of worm has 5 hearts to pump blood through it's long body. [Gilkes]

Many scientists consider information as the 5th dimension. [Luhn]

The number of seconds per hour divided by 5 factorial equals 5 primorial. [Luhn]

Euclid gave 5 postulates of plane geometry. [McCranie]

The American 5-cent piece called a "nickel" weighs 5.000 grams. It used to be made of nickel but is now mostly a copper alloy. [Lee]

5 is unique in that it is the sum of all primes less than 5. [Luhn]

The most cars have room for 5 people. [Luhn]

Some giraffe species have five horns on their heads. [Poo Sung]

Alan Turing's Erdős number. [Croll]

USA, UK, France, China, and Russia are 5 permanent members of the UN Security Council. [Poo Sung]

Mozart mastered his first musical composition before the age of 5. [Tallarico]

The only prime which is the difference of two prime squares. [Gupta]

The only prime sum of two consecutive Sophie Germain primes. [Silva]

The smallest safe prime. [Russo]

The smallest good prime. [Russo]

There are currently (August 2002) 5 known Mersenne primes with a prime number of digits. [Rupinski]

5 is the smallest first member of a twin primes pair (a, b) such that (a + b) +/- 1 is a second twin primes pair (c, d). [De Geest]

If you sum squares like 0^2 + 1^2 = 1, 0^2 + 1^2 + 2^2 = 5, 0^2 + 1^2 + 2^2 + 3^2 = 14, etc., it can be proven that the only prime you ever get for a sum is 5. [Burns]

There are 5 vowels in the English language: a, e, i, o and u. [Wagler]

Is "abstemious" the only English word which uses all 5 vowels just once in alphabetical order and contains the same number of consonants? [Bown]

5! + 5 = 5^{3}. [Woep]

Each 2^{2^n}-5 with n>3 cannot be the sum of two prime powers. [Zhi-Wei Sun]

The only prime digit in which a perfect square can end. [Gupta]

Several of *The Prime Pages* have been peer reviewed by the staff at MERLOT (Multimedia Educational Resource for Learning and Online Teaching) and rated 5 out of 5 in all categories.

The smallest prime comparable to all minimal primes in base 2. [Rupinski]

The first 5 open-end aliquot sequences are the so-called "Lehmer five."

5 is unique in that it is the only odd integer that can be expressed as the sum of two consecutive primes. [King]

The sum of all minimal primes in base 2 is prime. [Rupinski]

The largest minimal prime in base 4. [Rupinski]

F_{n} = 2^(2^*n*) + 1 are semiprimes for *n* = 5, 6, 7, and 8. [De Geest and
Honaker]

5 is unique in that it is the first and only prime with 5 as the last digit. [Gust]

Ackerman's function A(m,n) is never prime for any m ≥ 5. [Hartley]

The 5 Olympic rings overlap to create 3^2 separate areas. Note the first three primes in reverse order. [Miglinci]

5 stays in the middle of every magic square of order 3. Note that the magical sum for such a square is 5*3. [Laurv]

There are 5 books in the Bible with exactly 5 chapters. [Hartley]

5 is the only prime number surrounded by four prime factors (2*2, 5, 2*3). [Wagler]

The smallest number of queens needed to attack every square on a standard chess board. [Patterson]

Phi can be expressed as (5^.5*.5+.5). Perhaps 5 should be spelled *phive*. [Patterson]

!5 - 1 is prime. Note that !5 represents subfactorial 5. [Gupta]

5 = 12 in base 3. There is no other prime p that can be expressed as p = 12...n in base n+1, because p has n as a factor if n is odd, or n/2, if n is even. [Necula]

The smallest odd prime Thabit number. Arab mathematician Thabit ibn Qurra discovered a rule for finding amicable pairs based on these numbers (of the form 3 * 2^{n} - 1) before his death in Baghdad in 901.

An ace is a military aircraft pilot who has destroyed 5 or more enemy aircraft. [Richthofen]

5 is the answer to the question asked at the very end of the mathematics quiz show in the movie *Little Man Tate*.

The only sexy prime quintuplet starts at 5. I.e, 5+6, 5+6+6, 5+6+6+6 and 5+6+6+6+6 are also prime. [Opao]

The only single-digit prime with a prime ASCII code: 35 (in base 16) = 53. [Necula]

One of the best-known perfumes, *Chanel N°5*, was introduced by Gabrielle "Coco" Chanel on May 5, 1921.

"e" is the fifth letter of the English alphabet. [Dunham]

5 is the smallest number that appear in two twin Pythagorean triplets: (3, 4, 5) and (5, 12, 13). [Laurv]

5 is the smallest hypotenuse of a Pythagorean triangle (3^2 + 4^2 = 5^2). [Laurv]

5 is conjectured to be the only odd untouchable number. [Adams-Watters]

In esoteric writings, 5 is the number of magic that travels through cycles of four (e.g., seasons). [Woep]

The length of the side of a basic pantactic square (see Brian Astle, "Pantactic Squares" Math Gazette 1965, pp.144-52). [Astle]

Henry Ford is said to have paid his assembly-line workers an unheard of $5.00 a day, in 1914. [McAlee]

A Star Fish has 5 pointy legs. [McAlee]

All the solutions of n = π(2n) are nonnegative integers less than 5. [Firoozbakht]

5 is the first Centered Square Prime Number. [Post]

The first prime Apéry number. [Post]

5 is the smallest prime numerator of a Bernoulli number. [Post]

The only prime that is sandwiched between two semiprimes. [Gupta]

5 is the smallest nontrivial Euler number. [Terr]

"All number of the form n^4+4 , except 5, is the product of two integers." [Germain]

Evariste Galois was the first to show that equations of degree 5 cannot be solved by radicals. [Beedassy]

Divisibility test for 5: A number is divisible by 5 if it ends in 0 or 5.

A mathematical analysis says that our universe may well be a 5D black hole (*General Relativity and Gravitation*, Volume 37, p.1339).

The masculine marriage number to the Pythagoreans, uniting the first female number and the first male number by addition.

There are 5 families of butterflies. [Capelle]

5 is written with 5 lines on a calculator.

FIVE is the only prime written with an equal (prime) number of distinct vowels and consonants, i.e., IE & FV. [Beedassy]

Eugen J. Ionascu and collaborators were the first to establish that there can be no more than 5 Heron triangles (i.e., integer-sided triangles with integral areas) with two fixed prime sides. [Beedassy]

The only temperatures that are prime integers in both Celsius and Fahrenheit are +/- 5 °C (41 °F and 23 °F).

The only prime member of a Ruth-Aaron pair. [Caldwell]

The sum of the reciprocals of the primes is infinite, but
the sum of the reciprocals of the known primes is less than
5 *and will always be so!* [Caldwell]

Every positive integer can be written as
*x*^{2}+2*y*^{2}+7*z*^{2}+11*w*^{2}, except for 5. [Halmos1938] [Caldwell]

No perfect Golomb ruler exists for 5 or more ticks.

The difference between the number of primes less than ten thousand (1229) and the number of twin prime pairs less than one hundred thousand (1224).

Most humans have 5 fingers in the hands and feet. [Alvino]

The only prime p such that p divides the p^th Fibonacci number. All other primes divide either the (p-1)^st Fibonacci number if p = 1,4 mod 5 or (p+1)^st Fibonacci number if p = 2,3 mod 5. [Rupinski]

"Facetious" contains all 5 vowels of the English language in alphabetical order. [Giberson]

Adams was the first with a prime number of letters in his last name, among the Presidents of the United States of America. Obama is the most recent. [Post]

There is one chance in 5 for a random number on the number line to be an evil number. [Beedassy]

Boron has 5 letters and 5 protons in its atomic nucleus.

5 is the only prime in the middle of a pair of cousin primes. [Silva]

For the first five primes we have prime(2)*prime(4) - prime(1)*prime(3) = prime(5). It's interesting that we also have 2*4 - 1*3 = 5. [Firoozbakht]

The largest prime that can be the number of diagonals of an n-sided polygon (a pentagon in this case). [Green]

The only prime that is equidistant from a pair of primes by a prime (2) and also from a pair of nonprimes by a nonprime (4). [Beedassy]

The arithmetic average of the first sixteen digits of the decimal expansion of π. [Silva]

There are only 5 known regular polygons that can be constructed with Euclidean methods (i.e., by an unmarked straightedge and compass) that have a prime number of sides. [Green]

5 = 0^{12} + 0^{21} +
1^{02} + 1^{20} +
2^{01} + 2^{10} [Poo Sung]

The first number to appear between composites. [Silva]

The smallest number n such that neither n!+1 nor n!-1 is prime. [Silva]

The number of known strobogrammatic squares. The number of distinct digits to write them. [Capelle]

The largest Pell number (coincidentally prime) of the
form x^{2} + 1. Note that 5 =
2^{2} + 1, where 1 and 2 are the
two other Pell numbers of this form. [Capelle]

The only prime of the form 4^{n} + n^{4}, where n is a positive
integer. [Capelle]

Every positive integer can be written as the sum of 5 pentagonal numbers. Note that 5 is the only prime pentagonal number. [Capelle]

The only known prime p such that sigma(p) divides sigma(sigma(p)). [Capelle]

The only prime p such that phi(p) = tau(p) + 2, phi(p) = tau(p) * 2, and phi(p) = tau(p)^{2}. [Capelle]

The only prime equal to a prime p plus the pth prime. [Silva]

The largest known prime p such that fib(p) divides p!, but the only prime equal to fib(p). [Capelle]

There are 5 natural numbers n such that n is equal to the number of 5's in the decimal digits of all natural numbers smaller or equal to n. [Capelle]

Pollock's
conjecture (1850) states that every number can be written as the sum of at most five tetrahedral
numbers. No connection with the famous painting *No.
5, 1948*, by Jackson Pollock. [Capelle]

The smallest prime *p* such that both 2*p* + 1 and
2^{p} - 1 (= *M*_{p})
are primes. Note that both 2*M*_{p} - 1
and 2^{Mp} - 1 are also
primes. [Beedassy]

The largest known prime number p such that binomial(2p,p) is cubefree. [Capelle]

The only prime that always ends in itself when raised to any power. [Gundrum]

The only prime in mathematical physicist John Baez's personal list of favorite numbers.

The only greater of twin primes, written in base 6, not to end with the digit 1. Because all but the first greater of twin primes is of the form 6n+1, hence ends with (base 6) the digit 1. [Post]

There are 5 known repunit primes. [Green]

The smallest prime *p* such that *p*! + 1 is the
square of a prime with a prime number of digits. [Beedassy]

There are 5 topological symmetry types for oriented knots: fully symmetric, reversible, +amphicheiral, -amphicheiral, and asymmetric. Not to be confused with prime knots. [Post]

The only prime p that is the sum of the prime divisors of p+1. [Silva]

The only prime whose the square is composed of only prime digits. [Loungrides]

The primes which can be written with 5 binary digits have a total of 2^5 nonzero binary digits. [Hasler]

Both (prime(5))!/5!+1 and (prime(5!))!/(5!)!+1 are prime. [Merickel]

The only Fibonacci prime that is the sum of two others. [Silva]

π(1) + π(2) + π(3) + π(4) + π(5) = phi(1 + 2 + 3 + 4 + 5). It is the only number with this property. [Firoozbakht]

The floor function of e^phi = 5, where phi is golden ratio. [Gupta]

The concept of a regular polyhedron of index two was introduced in Wills for orientable polyhedra with planar faces. There are exactly 5 such polyhedra. [Post]

The minimum number of Pythagorean triangles required to tile a square (C. Jepsen and R. Yang). [Beedassy]

Canada/USA Mathcamp is an intensive 5-week-long summer program for mathematically talented high school students.

5 is the initial prime number which is not a factor of any member of the Lucas sequence. [Schiffman]

5 primorial falls between the 5th set of twin primes. [Merickel]

a(5) of Sloane A177876 is unknown. It will require the factorization of an 18742-digit integer.

"Strengths" is the only English word to contain 5 consecutive consonants. [Homewood]

'A' is the only letter in the English alphabet that contains 5 non-straight angles (three acute and two obtuse). [Green]

The end of a nucleic acid where the phosphate group is located is called the 5 prime end. [Chin]

*Five-number summary* is another name for the visual
representations of the box-and-whisker plot. [Nord]

5 is the first prime position in Grether's Spiral.

The largest known number (coincidentally prime) n such that n! is a triangular number. [Gupta]

A revised paper by Terence Tao is titled 'Every odd number greater than one is the sum of at most 5 primes'. [Beedassy]

*Cri-du-chat
syndrome* is a rare genetic disorder caused by an abnormality in chromosome number 5. [Beedassy]

Deoxyribose is a 5-carbon carbohydrate.

All (prime(n+1)/prime(n))^n values, with n greater than or equal to 5, are less than n*log(n). [Nicholson]

The first term in the only sexy prime quintuplet, (5, 11, 17, 23, 29). Note that 112317529 is the smallest prime formed by concatenating the terms of this quintuplet. [Loungrides]

The only known prime of form 2^p+1, where p is prime, (case p=2). [Loungrides]

5!-7 and 5!+7 give two consecutive primes, whereas 5!-11 and 5!+11 give the previous and the next consecutive primes. [Petrov]

SOUTH AMERICA is the only continent with all 5 vowels in its name. [Gupta]

The only prime automorphic (also called curious) number, i.e., n^2 ends with n.

Trudgian conjectures there are no primes p > 5 for which 2!,3!, ... ,(p−1)! are all distinct modulo p.

5! + 4! - 3! + 2! - 1! and 5! - 4! + 3! - 2! + 1! are each members of a twin prime pair! [Homewood]

Using one each of 2^a and 3^b, 5 = 1+4 = 2+3 = 8-3 = 9-4 = 32-27 for five different ways. [Bergot]

The only Cyclops prime in binary [Homewood]

There are normally 5 petals in the family *Rosaceae*.

In chess, the "rule of 5" is for positions in which the pawn is protected and the opposing king is cut off by files: Add the number of rank of the pawn to the number of files the defender's king is cut off. If the sum is more than five, it is usually a win. Otherwise it is normally a draw. [Soltis and Mednis]

The first upside-down number. [Kennedy and Cooper]

Every Rhonda number to base ten contains at least one 5. [Zumkeller]

Hamming numbers (also known as ugly numbers) are numbers whose prime divisors are less or equal to 5.

“Detective Sanders and the Zodiac Killer” is a Puzzle Crime
5tory, exclusively created for *Puzzle Prime.*

6*5+1, 66*5+1, 666*5+1, 6666*5+1, 66666*5+1, 666666*5+1 and 6666666*5+1 are all primes. [Sariyar]

Helfgott proved that every odd number larger than 5 is the sum of three primes.

There are 5 prime quintuplets of the form {p, p + 4, p + 6, p + 10, p + 12} less than 10^4. [Homewood]

(5, 7, 11, 13) is the first tetrad of successive primes p < q < r < s such that (p+q) divides r*s+1, i.e., (5+7) divides 11*13+1. The next such tetrad is (47, 53, 59, 61). Is there other such tetrad? [Loungrides]

Amateur mathematician Aubrey D.N.J. de Grey constructed a unit-distance graph which cannot be colored with fewer than 5 colors.

"Queueing" is the only English word to contain 5 consecutive vowels. [Homewood]

5 is the last digit to appear in decimal expansion of Golden ratio. [Gupta]

5 months will all have 5 Thursdays in 2020 (i.e., January, April, July, October and December). Note that Thursday is the 5th day of the week! [Reynolds]