# 65

This number is a composite.

The first 65 digits of 65^{65} form a prime number. [Honaker]

The product of the first two primes of the form 4*n* + 1.

Euler found 65 integers, which he called "numeri idonei," that could be used to prove the primality of certain numbers. [Brown]

6^{65} - 5 is the smallest prime of the form *a*^{c} - *b*, where
*b* = *a* - 1. Note that *c* is the concatenation of *a* and *b*. [Kulsha]

The smallest composite number of the form *n*^{2} + 1, where *n* is even.

65 is the ONLY number which gives a prime square by adding as well as subtracting its reversal from it: 65 + 56 = 11^{2}, 65 - 56 = 3^{2}. [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]

The smallest Fermat semiprime. [Capelle]

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. You can 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]