# 3

This number is a prime.

Jason Funk © 2002 |

The smallest reflectable prime.

The Italian-born French mathematician Joseph-Louis Lagrange (1736-1813) spent much of his life working on the 3-Body Problem.

The first in a pair of primes of the form (*p*, *p* + 4) called cousin primes.

The smallest odd Fibonacci prime. It is the only Fibonacci prime with a composite index number: 3 = fib(4).

A mark on a small circle, rolling inside one with three times the diameter, traces out a 3-cusped hypocycloid. Euler called it a deltoid because of its resemblance to the Greek letter delta.

2^{3} ± 3 are primes. [Brown]

The integer part of 3^{33}/33^{3} is prime! [Kulsha]

Choose a prime number greater than 3. Multiply it by itself and add 14. If the result is divided by 12, then the remainder will always be 3.

If N = *p*^{3}, where *p* is prime, then *N* is multiplicatively perfect. [Gudder]

The smallest Fortunate number.

Captain Kirk and Spock played chess 3 times on the television series *Star Trek*. Kirk won every game.

Vinogradov's theorem states that every sufficiently large odd integer is a sum of at most 3 primes.

Divisibility test for 3: A number is divisible by 3 if the sum of its digits is divisible by 3. (The same is true for nine.) [Greene]

The smallest Fermat prime.

Every Carmichael number is the product of at least 3 distinct primes. [Menezes , Oorschot and Vanstone]

The "Three-fold Law" is a common tenet held by some Wiccans stating that both the good and the evil that one creates in the world come back to benefit or hurt them -- magnified 3 times over.

Racing legend Dale Earnhardt drove the number 3 car for most of his career. (His first car was pink "K-2".)

In base three the only permutable primes are 12 and 21.

3^{3} = 3^{2} + 3^{2} + 3^{2}. [Rivera]

Nicola Tesla (1856--1943), inventor, electrical engineer, and physicist, was obsessed with the number 3. For example, it was not uncommon to see him walk around a block 3 times before entering a building.

The *Associated Press* once reported that a dog was born in Gallatin, Tennessee, with the number 3 on its ears.

An *odd amicable pair* with only one member divisible by 3 has not been found.

Sharkovsky's theorem states that if a continuous real-valued function has a point of period 3 (i.e., x=f(f(f(x)))), then it has points of every other period.

We all know that 3^{2} + 4^{2} = 5^{2} ... but less well-known is 3^{3} + 4^{3} + 5^{3} = 6^{3}. [Goldstein]

The only prime, *p*, such that *p* + 1 is a square.

If White's chess pieces are on their original squares and Black has only a king on h4, then White can checkmate Black in 3 moves. [Loyd]

The only triangular number which is prime. [Gupta]

The 3-toed sloth reaches sexual maturity at about 3 years of age. [Jinsuk]

The number of bits required to represent an instruction in BF, the minimalist eight-instruction Turing-complete computer programming language. [Croll]

The True Prime: "If I tell you 3 times, it's true." -- Lewis Carroll

We live on the 3rd planet from the Sun. [Levin]

3 is the only number which is equal to the sum of all the natural numbers less than it. [Murthy]

The dog-sized *Eohippus* ("dawn horse") had 3 hoofed toes on each hind foot. [Marsh]

The German card game Skat requires at least 3 players. [Luhn]

The only prime whose factorial is a perfect number. [Murthy]

The largely self-taught Indian genius Srinivasa Ramanujan (1887-1920) proposed that sqrt(1 + sqrt(1 + 2*sqrt(1 + 3*sqrt(1 + 4*sqrt(1 + ...))))) = 3.

The only prime sandwiched between a prime and a composite number. [Murthy]

Aulacogens are the failed rifts of 3-armed rift systems.

3 is the only prime *p* such that *p*^{2} + 2 is prime. [Luhn]

The only Fermat number which is also a triangular number. [Gupta]

The only Fibonacci prime that is also a triangular number. [Russo]

The only prime *p*, such that *p* = π(*p*!). [Firoozbakht]

The smallest prime *p* such that the previous prime concatenated with the next prime forms a square (25). [De Geest]

The 3 periods of the Mesozoic Era were the Triassic, the Jurassic, and the Cretaceous. [Dobb]

On Earth, water exists naturally in all 3 phases; solid, liquid and gas. [Wagler]

3 is the only integer *n* such that *n*!+1 and *n*!-1 are both primes. [Gupta]

There are 3 operations in the encryption process for the Even-Mansour cipher: an initial exclusive or, a permutation and a final exclusive or. [Croll]

In the fifth and all higher dimensions there are only 3 regular polytopes. [Rupinski]

There are 3 additive primary colors (red, green, and blue) and 3 subtractive primary colors (cyan, magenta, and yellow).

(2^(p-1)-1)/p is congruent to 0 (mod 3), for all primes p greater than 3. This is a corollary to Fermat's Little Theorem. [Dunn]

3 is the only positive integer n which is the difference between the least prime greater than n and the greatest prime less than n. [Hartley]

Terza rima is a verse form composed of iambic tercets (3-line groupings). [Palmer]

3! + 3 = 3^{2}. [Woep]

2^{2^1}-1=3; 2^{2^2}-1 can be written as the sum of a prime power and two distinct powers of two in exactly 3 ways (namely, 2+2^2+3^2, 2^2+2^3+3, 2+2^3+5); 2^{2^3}-1=2^2+2^3+3^5=2+2^7+5^3. For any integer n>3, 2^{2^n}-1 cannot be expressed as the sum of a prime power and two distinct powers of two. [Zhi-Wei Sun]

DSL Prime news is distributed 3 times a month.

Only one rare number ending in 3 has ever been found, i.e., 8888070771864228883913. [Gupta]

The greatest number of consecutive integers which are pairwise relatively prime. [Rupinski]

As at February 2003, 3 and 13 are the only *known* prime numbers n for which the number of n-ominoes is also prime. [Hartley]

The Pythagoreans considered 3 to be the first masculine number.

The function *n*^(1/*n*) achieves its maximum value for integers *n* at *n* = 3. [Rupinski]

According to the quadratic reciprocity theorem, if p and q are distinct primes of the form 4k+3, then exactly one of the following congruences is solvable in integers: x^2 = p mod q or x^2 = q mod p. [Rupinski]

3 and 103 are prime, as are 3*2^103-1 and 103*2^3-1. There is no other pair of primes less than 1000 with this property. [Hartley]

The only Sophie Germain Prime p of the form 4k+3 such that M_{p} is prime. All larger M_{p} for such p are composite as shown by Euler and Lagrange. [Rupinski]

3 is the only time that 2/3rds or 66.6....% of numbers to that point are prime. [Gust]

3^2^1 equals 3!+2!+1! [Silva]

3 is the earliest prime p such that p^5 is the sum of 5 consecutive primes: 3^5 = 41 + 43 + 47 + 53 + 59. [Rivera]

The logo of Hutchison 3G is prime. [Patterson]

*NUMB3RS* is an American television show that airs on CBS. In one episode called *Prime Suspect*, a young girl's kidnapping is related to her father's work on the Riemann hypothesis.

3 is the smallest prime such that if F(p) = 2*p^2-1, then F(3), F(F(3)), and F(F(F(3))) are all primes. [Opao]

The smallest possible Proth prime, *k**2^{n}+1 with *k* < 2^{n}, here *k*=1, *n*=1. [Rupinski]

The only number (curiously prime) whose subfactorial is also prime. [Gupta]

3 is the value of π calculated from the measurements in I Kings 7:23 (if we assume the measurements are not approximations and the measurements both refer to the outside edge of the vessel). [Silva]

!3 + 1 is prime. Note that !3 represents subfactorial 3. [Gupta]

The only twin prime triplet starts at 3. I.e., 3+2 and 3+2+2 are also prime. [Opao]

The only cousin prime triplet starts at 3. I.e., 3+4 and 3+4+4 are also prime. [Opao]

In most jurisdictions, a tablespoon equals 3 teaspoons (but it is 2 teaspoons in Asia and 4 in Australia).

3 is the first Mersenne prime. [Rajh]

A codon is a sequence of three adjacent nucleotides, which codes for an amino acid. [Necula]

Texas state law prohibits taking more than 3 sips of beer at a time while standing. [Patterson]

1!*2!*3! ± 1 are twin primes. [Gupta]

Among integers, the prime number 3 is the best approximate for three important transcendental numbers: π, Napier's number and the Gelfond-Schneider constant. [Ranta]

3 is the only prime number with a digital root of 3. [McAlee]

There are only 3 non-trivial pseudoprimes less than a hundred (49, 77, 91). [McAlee]

There are only 3 irregular primes less than 100: 37, 59, 67. [McAlee]

3#!-1 is prime. [Patterson]

π(3) = !3, where !3 denotes subfactorial 3. [Gupta]

A quaternion depends on 3 imaginary units. [Croll]

The smallest prime of the form x^2 - y^2, where x and y are two consecutive numbers. [Teofilatto]

3^2, 3^3, 3^4, and 3^5 have the same sum of digits. [Gallardo]

3 is the first prime Delannoy number. [Post]

There exists 3 and only 3 groups (or generations) of quark couples as well as those of lepton couples that comprise the entire family of elementary particles of matter known as fermions. [Beedassy]

The smallest Gaussian prime. [Beedassy]

The smallest prime with a prime subscript. [Beedassy]

The law of proportions, called "Rule of Three" by the Indian mathematician Brahmagupta (598-668), became a standard of rational thought. For example, Abraham Lincoln wrote that as a young man he "could read, write, and cipher to the Rule of Three." Charles Darwin wrote "I have no faith in anything short of actual measurement and the Rule of Three." Perhaps less known is the fact that they were born on the exact same day (February 12, 1809).

The smallest triadic prime. [Capelle]

The only prime number adjacent to a prime number in any row of Pascal's Triangle. [Green]

If n is any odd integer > 1, and x is any positive integer, then x^n - x is a multiple of 3. [Lee]

The smallest prime with prime period continued fraction for its square root. [Beedassy]

The only prime that cannot be the arithmetic average of two primes. [Silva]

The only 'unwanted' number, as in "three's a crowd!"

The only value of p such that p and p + 2 are twin primes and p(p + 2) + p - 1 and p(p + 2) + p + 1 are also twin primes. [Opao]

The terms of the sequence 3/2, (5 + 7)/(2 + 3), (7 + 11 + 13)/(2 + 3 + 5), (11 + 13 + 17 + 19)/(2 + 3 + 5 + 7), etc., converge to 3 as the primes used approach infinity.

There are 3 sets of chromosomes in the mosquito. [Anopheles]

The smallest prime *p* such that
π(*p*^{2}) = (π(*p*))^{2}. [Capelle]

The only odd prime that is a highly abundant number. [Beedassy]

The National Basketball Association adopted the 3-point field goal in 1979. [Lee]

The first lucky prime. Since lucky numbers are lucky enough to repeatedly appear in this book, let's take a moment to define them. Start with the list of natural numbers: 1, 2, 3, ... , and cross out every second number. The second number not crossed out is 3, so we cross out every third number, leaving 1, 3, 7, 9, 13, 15, .... The third number left is 7, so we cross out every seventh number---repeat forever. What remains is the sequence of lucky numbers: 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, .... S. Ulam (1909-1984) investigated these numbers and found a strong resemblance to primes.

The only prime *p* such that (*p*^{5} - 1)/(*p*
- 1) is the square of a prime. [Beedassy]

(a^(p)-a)/p is congruent to 0 (mod 3), for all primes p greater than the positive integer a, except where p = 3, a = 2. [Papadimitriou]

The Fermat point of a triangle is located at a minimum total distance from 3 other points.

The only prime of the form n^2 - 1. [Pimentel]

The smallest prime p such that 2^p + p is prime. [Silva]

The only prime *p* whose factorial is the sum of the first *p*
numbers. [Silva]

The first prime of the form p=2*n^n+1. [Loungrides]

The number of sides of the only regular polygon which cannot be formed with n-3 folds of a rectangular strip of paper where n is the number of sides of the polygon [Fellows]

The only number that is both an odd prime number and a highly composite odd number. [Beedassy]

The only state capital containing 3 words is Salt Lake City.

There are three prime odd digits. [Silva]

It is conjectured that the Erdős-Moser equation
1^{k}+2^{k}+...+(m-1)^{k}
= m^{k} has an integer solution
only for m = 3: 1^{1} +
2^{1} =
3^{1}. [Capelle]

The only prime p whose double factorial (p!!) has p digits. [Silva]

The only prime p whose square is the sum of the factorials of the first p numbers. [Silva]

The smallest odd prime with a prime number of partitions. [Pol]

The only prime, p, such that 2^{p} is congruent
to -1 (mod p). [Wesolowski]

The only prime, p, such that σ(p^{4}) is a square. [Wesolowski]

The only Wagstaff prime that is equal to the exponent of the form that creates it. [Loungrides]

According to Hindu scriptures the Universe is a
3-world-cosmology abode called *"Triloka"* that
comprises the perceptible Physical world along with the
subtler realms of the Celestial and the Nether worlds. [Beedassy]

Uniquely, 3 is the only prime whose powers (greater than 1) all have a digital root which is its own square. The simple proof is a direct consequence of the divisibility rule. [King]

The minimum number of moves required to solve the Turn the Fish Puzzle. [Lester]

The only prime p such that phi(p) is prime. [Firoozbakht]

The 3rd power of 3 equals 3 squared plus the sum of the first 3 composite numbers. [Silva]

"Houray !" exclaimed aloud 3 times in response to "Three cheers for ...": "Hip Hip Hip !" (uttered 3 times in succession) is a common jubilatory and glorifying interjection during social gatherings in many British-impregnated cultures. [Beedassy]

The floor function of phi^e = 3, where phi is golden ratio. [Gupta]

Popcorn kernals can pop up to 3 feet in the air!

The number of primes less than the first perfect number.

Scruggs (or 3-finger) style is the most common style of playing the banjo in bluegrass music.

April "3 cubed" is National Prime Rib Day.

The Christian doctrine of the Trinity holds that God exists as 3 Divine Persons, i.e., the Father, the Son (Jesus Christ), and the Holy Spirit. [Silva]

It is not known whether a dissection of an equilateral triangle to a square is possible with 3 pieces.

The area under a cycloid is 3 times the area of the generating circle. [Beedassy]

Camels have 3 eyelids over each eye. [Northcutt]

Kannan Soundararajan and Robert J. Lemke Oliver of Stanford University found that primes ending in 3 seem to like being followed by primes ending in 9 more than 1 or 7. Prime conspiracy?

The Erdős–Kac theorem means that the construction of a number around one billion requires on average 3 primes.

The smallest number in the only triad of consecutive odd prime integers. [Silva]

Find the unique solution of the following cryptarithmic addition (proposed by Sidney Kravitz): OTTAWA + CALGARY = TORONTO. Hint: The sum of digits of all 3 numbers is the smallest 3-digit number.

The end of a nucleic acid where the sugar group is located is called the 3 prime end. [Chin]

Can you find 3 cases such that p^2 + q^2 = r^2 + 1, where p, q, r are consecutive primes? [Bergot]

The only prime Pisano period. [Homewood]

Triduum is a 3-day period of religious observance in the Catholic Church. [Francis]

In a paper dated 19 Feb 2016, authors J. Cilleruelo, F. Luca, L. Baxter proved any positive integer is the sum of 3 palindromes in base 5 or beyond. [Beedassy]

Baryons are made of an odd number of quarks - usually 3.

3 consecutive tau numbers do not exist. [Zelinsky]

"Trey" is a cardplayer's term for 3. [Demuth]

The sum of prime numbers up to 3 is the 3rd prime. [Wiszowaty]

Paul Erdős observed that, when n is a prime number in the no-three-in-line problem, the set of n grid points (i, i^2 mod n), for 0 ≤ i < n, contains no 3 collinear points.

Sitar is a Persian word meaning *3 strings*. [Caudhurī]

"Thirdsday" comes once every seven years or so, whenever January 3 falls on a Thursday. [Beedassy]

3 = 569936821221962380720^3 + (-569936821113563493509)^3 + (-472715493453327032)^3. [Booker]

The smallest periodic binary prime - 11 in binary. [Croll]

The first systematic study of Bell numbers was made by Ramanujan in Chapter 3 of his second notebook. [Becker]

Rubella (aka 3-day measles) can cause complications during pregnancy. [Combs]

The product of the smaller in a set of twin primes (3), the next integer (4), and the larger twin (5), equals 60, which falls between another set of twin primes {59, 61}. In other words, {p, p+2} and {p*(p+1)*(p+2)-1, p*(p+1)*(p+2)+1} are two sets of twin primes, where p is prime (smallest case p=3), i.e., {3,5} and {59,61}. Can you find the smallest prime p where this occurs recursively for 3 sets? 4 sets? 5 sets? etc. The sequence begins 3, 53137619, 2856646544865959, ... . [Honaker]

Only prime p that divides R_{p}, where R_{p} denotes repunit with prime subscript p. [Gupta]