Lucky numbers of Euler


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.
Leonhard Euler published the polynomial 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.
The primes of the form k2 − k + 41 are
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.

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