Lucky numbers of Euler

Euler's "lucky" numbers are positive integers n such that m² − m + n is a prime number for m = 0, …, n − 1.

Leonhard Euler published the polynomial x² − x + 41 which produces prime numbers for all integer values of x from 0 to 40. Obviously, when x is equal to 41, the value cannot be prime anymore since it is divisible by 41. Only 6 numbers have this property, namely 2, 3, 5, 11, 17 and 41 (sequence A014556 in OEIS).

These numbers are not related to the so-called lucky numbers.

See also

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

References

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

External links

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