# 65

This number is a composite. The first 65 digits of 6565 form a prime number. [Honaker] The product of the first two primes of the form 4n + 1. Euler found 65 integers, which he called "numeri idonei," that could be used to prove the primality of certain numbers. [Brown] 665 - 5 is the smallest prime of the form ac - b, where b = a - 1. Note that c is the concatenation of a and b. [Kulsha] The smallest composite number of the form n2 + 1, where n is even. 65 is the ONLY number which gives a prime square, on adding as well as subtracting its reverse from it (65 + 56 = 112, 65 - 56 = 32). [Gupta] The only number which is the difference of fourth powers of two primes. [Murthy] (65!)2 + 1 is prime. [Dobb] 65 = 5 * 13 and π(65) = π(5) * π(13) = 5 + 13. [Honaker] π(65) = phi(65) - 6*5. [Langroudi] The smallest square-free semiprime whose ternary representation (2102) is also semiprime when read in decimal. [Bajpai] The reversal of 65^56 is prime.

Copy and paste RevDigits(65^RevDigits(65,10),10) into the 'Value space' at Alpertron (integer factorization calculator) then click on the 'Prime box' to display this 102-digit number. Can you find a larger prime of this form? [Honaker]