Euler's "lucky" numbers are positive integers n such that for all integers k with , the polynomial produces a prime number. Characteristics 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. As established by Kurt Heegner in 1952, 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 k2 − k + 41 are 41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 853, 911, 971, ... .See also the sieve algorithm for all such primes: Other lucky numbers 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.
See also Heegner number List of topics named after Leonhard Euler Formula for primes Ulam spiral
References
Literature Le Lionnais, F. Les Nombres Remarquables. Paris: Hermann, pp. 88 and 144, 1983. Leonhard Euler, Extrait d'un lettre de M. Euler le pere à M. Bernoulli concernant le Mémoire imprimé parmi ceux de 1771, p. 318 (1774). Euler Archive - All Works. 461.
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