5

This number is a prime.

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5 is the smallest prime p = Prime(n) such that Prime(n) + Prime(n+p) = Prime(n+1) + Prime(n+p-1) = Prime(n+2) + Prime(n+p-2). There are 5 more such primes less than 1000. [Ahmed]

The only n such that n! + 1 = pₙ². [Ahmed]

5 is the only prime p for which there exist positive integers a, b such that p = a + b = prime(a) - prime(b): 5 = 4 + 1 = prime(4) - prime(1) = 7 - 2. [Ahmed]

5 is the only known prime of the form σₙ(n), the sum of the n-th powers of the divisors of n. (See OEIS A023887) [Ahmed]