Prime Curios!: 4

4
(another Prime Pages' Curiosity)

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(!4) = 4, where !4 denotes subfactorial 4. [Gupta]

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 4x4 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 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 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 4s 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]