 (!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]
4!4 + 1 is prime.
18 99 86 61
66 81 98 19
91 16 69 88
89 68 11 96
A 4-by-4 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! ... (p-1)!]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):
* * *
x * *
---------
* * * *
* * * *
-----------
* * * * *
There is only one solution. [Mensa]
The Lucas-Lehmer Test states that for p odd, the Mersenne number
2p - 1 is prime iff 2p - 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 exactly 4. [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]
44 - 4! + 1 is prime. [Luhn]
Smallest composite number such that the sum of proper divisors is a prime. [Russo]
There are 4 two-digit primes formed by concatenation of
prime digits. [Silva]
4 is the exact number of one-digit 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]
4 and (4 + 1) are the only consecutive numbers n , where n!+ 1 a square prime number. [Luhn]
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 105, such that p = 2n + 3n 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*4-4-4/4. [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 4-digit 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]
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