# Lucky numbers of Euler

Euler's "lucky" numbers are positive integers n such that for all integers k with 1 ≤ k < n, the polynomial k2k + n produces a prime number.

When k is equal to n, the value cannot be prime since n2n + n = n2 is divisible by n. Since the polynomial can be written as k(k−1) + n, using the integers k with −(n−1) < k ≤ 0 produces the same set of numbers as 1 ≤ k < n.

Leonhard Euler published the polynomial k2k + 41 which produces prime numbers for all integer values of k from 1 to 40. Only 7 lucky numbers of Euler exist, namely 1, 2, 3, 5, 11, 17 and 41 (sequence A014556 in the OEIS).

The primes of the form k2k + 41 are

41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 853, 911, 971, ... (sequence A005846 in the OEIS).

The terminology is ambiguous: "Euler's lucky numbers" are neither the same as, neither related to the "lucky numbers" defined by a sieve algorithm. In fact, the only number which is both lucky and Euler-lucky is 3, since all other Euler-lucky numbers are congruent to 2 modulo 3, but no lucky numbers are congruent to 2 modulo 3.