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Lucky numbers of Euler
Euler's "lucky" numbers are positive integers n such that for all integers k with 1 ≤ k < n, the polynomial k2 − k + n produces a prime number.
When k is equal to n, the value cannot be prime since n2 − n + 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. These polynomials are all members of the larger set of prime generating polynomials.
Leonhard Euler published the polynomial k2 − k + 41 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 (sequence A014556 in the OEIS). Note that these numbers are all prime numbers.
The primes of the form k2 − 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.
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Lucky numbers of Euler
Euler's "lucky" numbers are positive integers n such that for all integers k with 1 ≤ k < n, the polynomial k2 − k + n produces a prime number.
When k is equal to n, the value cannot be prime since n2 − n + 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. These polynomials are all members of the larger set of prime generating polynomials.
Leonhard Euler published the polynomial k2 − k + 41 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 (sequence A014556 in the OEIS). Note that these numbers are all prime numbers.
The primes of the form k2 − 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.