Euler's "lucky" numbers are positive integers n such that for all integers k with , the polynomial produces a prime number.
When k is equal to n, the value cannot be prime since is divisible by n. Since the polynomial can be written as , using the integers k with produces the same set of numbers as . These polynomials are all members of the larger set of prime generating polynomials.
Leonhard Euler published the polynomial which produces prime numbers for all integer values of k from 1 to 40. Only 6 lucky numbers of Euler exist, namely 2, 3, 5, 11, 17 and 41 . Note that these numbers are all prime numbers.
The primes of the form k<sup>2</sup> â k + 41 are
Euler's lucky numbers are unrelated 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.