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(!4) = 4, where !4 denotes subfactorial 4. [Gupta] The smallest composite number. Every prime that is one more than a multiple of 4 can be written as the sum of two squares in precisely one way. [Fermat] 18 99 86 61 66 81 98 19 91 16 69 88 89 68 11 96A 4by4 magic square displaying 4 primes. Note that the same occurs if rotated 180 degrees about the center of the square's plane. (x) is greater than or equal to log(x)/log 4. [Erdös] 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ... + 1/p exceeds 4 when p is 1801241230056600523. [Bach and Sorenson] [2!3!4! ... (p1)!]^{4} is congruent to 1 (mod p), for p a prime. [Rados] Let n be an integer, and let C(n) be the number of ways n can be written as the sum of two primes. The numbers, C(n) are called Goldbach numbers. E.g., C(34) = 4 because 34 can be written as 3 + 31, 5 + 29, 11 + 23, and 17 + 17. No number of the form 4n  1 is a sum of two squares. The following cryptarithm utilizes the first 4 prime digits (2, 3, 5, & 7): There is only one solution. [Mensa]* * * x * *  * * * * * * * *  * * * * * The LucasLehmer Test states that for p odd, the Mersenne number 2^{p}  1 is prime iff 2^{p}  1 divides S(p  1) where S(n + 1) = S(n)^{2}  2, and S(1) = 4. The mean gap between successive primes up to 337 is the first composite number. [Honaker] 4 is the only square between twin primes. [Gundrum] The largest number such that its divisors + 1 are primes. [Murthy] In the movie "Die Hard with a Vengeance", Bruce Willis must measure 4 litres of water using two containers one with a capacity of 3 litres and the other with 5 litres. This problem can be solved by the Euclidean algorithm since 3 and 5 are mutually prime. [Poo Sung] 1!^{1} +2!^{2} + 3!^{3} + 4!^{4} is prime. [Poo Sung] 4^{4}  4! + 1 is prime. [Luhn] Smallest composite number such that the sum of proper divisors is a prime. [Russo] There are 4 twodigit primes formed by concatenation of prime digits. [Silva] 4 is the exact number of onedigit primes. [Patterson] Four fours raised to the fourth power plus one is prime. I.e., 4444^4+1 is prime. [Opao] 4 fives plus five 4's is prime. [Opao] If we add 4 and the 4th Fibonacci number (which is prime), we find the 4th Lucas number (which is the 4th prime number). [Capelle] If S(n) is the sum of the first n primes, then the limit of S(2n)/S(n) = 4, as n approaches infinity. [Capelle] The largest value for n less than 10^{5}, such that p = 2^{n} + 3^{n} is prime. [Opao] 4 is the largest number m such that prime(m) = sigma(m). [Firoozbakht] 4444^(4*4) + 4/4 is prime with prime length 4*4*444/4. [Memar] The smallest semiprime and also the smallest brilliant number. [Luhn] 4 is the smallest number n such that n and n! are product of factorials of primes (4 = 2!2! and 4! = 2!2!3!). [Capelle] 4 is the smallest number between twin primes. [Silva] The only semiprime expressed by a prime p in two ways: p^2 and p+p. [Silva] The number of 4digit primes formed from the first 4 odd digits. [Silva] The only square which is arithmetic average of a pair of twin primes. [Silva] 4!!3!!+2!!1!! is a 4!digit prime. [Silva] There are 4 known positive integers n such that the sum of all primes smaller or equal to n divides n(n+1)/2. This sum also divides the sum of all nonprimes smaller or equal to n. Note that 4 is the smallest positive integer with this property. [Capelle] The only known number n such that prime(n)^prime(n)  (n) is prime. [Firoozbakht] 4 is the number of primes formed from 4 digits summing up to 4 [Silva] The 4th prime that gives primes in translation twice from bases 2 and 3 to base 10 is the first to do it once from base 4. The 44th is the first to do it twice, and both begin with 234 in base ten (234099253 and 2348568403). [Merickel] (4!) = !4. [Wesolowski] The only digit that forms a prime with the previous digit (43). [Silva] No palindromic prime has a sum of digits equal to 4. [Green] The only known composite number n such that n^n+1 is a nontitanic prime. [Loungrides]
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