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.

Obviously, when k is equal to n, the value cannot be prime anymore since n² − n + n = n² is divisible by n. Since the polynomial can be rewritten 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 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).

These numbers are not related to the lucky numbers generated by a sieve algorithm.

See also

• Heegner number
• List of topics named after Leonhard Euler
• Formula for primes

References

• Le Lionnais, F. Les Nombres Remarquables. Paris: Hermann, pp. 88 and 144, 1983.

External links

• Weisstein, Eric W. "Lucky Number of Euler". MathWorld.