Lucky numbers of Euler
{{Short description|Mathematical concept}}
Euler's "lucky" numbers are positive integers n such that for all integers k with {{nowrap|1 ≤ k < n}}, the polynomial {{nowrap|k2 − k + n}} produces a prime number.
Characteristics
When k is equal to n, the value cannot be prime since {{nowrap|n2 − n + n {{=}} n2}} is divisible by n. Since the polynomial can be written as {{nowrap|k(k−1) + n}}, using the integers k with {{nowrap|−(n−1) < k ≤ 0}} produces the same set of numbers as {{nowrap|1 ≤ k < n}}. These polynomials are all members of the larger set of prime generating polynomials.
Leonhard Euler published the polynomial {{nowrap|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 {{OEIS|id=A014556}}.{{Cite web |last=Weisstein |first=Eric W. |title=Lucky Number of Euler |url=https://mathworld.wolfram.com/LuckyNumberofEuler.html |access-date=2024-09-21 |website=mathworld.wolfram.com |language=en}} 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, ... {{OEIS|id=A005846}}.See also the sieve algorithm for all such primes: {{OEIS|id=A330673}}
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
References
{{refs}}
Literature
- Le Lionnais, F. Les Nombres Remarquables. Paris: Hermann, pp. 88 and 144, 1983.
- Leonhard Euler, [https://scholarlycommons.pacific.edu/euler-works/461/ 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.
External links
- {{MathWorld|urlname=LuckyNumberofEuler|title=Lucky Number of Euler}}
{{Classes of natural numbers}}