In mathematics, Euler's idoneal numbers (also called suitable numbers or convenient numbers) are the positive integers D such that any integer expressible in only one way as x2 ± Dy2 (where x2 is relatively prime to Dy2) is a prime power or twice a prime power. In particular, a number that has two distinct representations as a sum of two squares is composite. Every idoneal number generates a set containing infinitely many primes and missing infinitely many other primes.

Definition

A positive integer n is idoneal if and only if it cannot be written as ab + bc + ac for distinct positive integers a, b, and c.[1]

It is sufficient to consider the set { n + k2 | 3 . k2ngcd (n, k) = 1 }; if all these numbers are of the form p, p2, 2 · p or 2s for some integer s, where p is a prime, then n is idoneal.[2]

Conjecturally complete listing

Unsolved problem in mathematics:

Are there 65, 66 or 67 idoneal numbers?

The 65 idoneal numbers found by Leonhard Euler and Carl Friedrich Gauss and conjectured to be the only such numbers are

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 18, 21, 22, 24, 25, 28, 30, 33, 37, 40, 42, 45, 48, 57, 58, 60, 70, 72, 78, 85, 88, 93, 102, 105, 112, 120, 130, 133, 165, 168, 177, 190, 210, 232, 240, 253, 273, 280, 312, 330, 345, 357, 385, 408, 462, 520, 760, 840, 1320, 1365, and 1848 (sequence A000926 in the OEIS).

Results of Peter J. Weinberger from 1973[3] imply that at most two other idoneal numbers exist, and that the list above is complete if the generalized Riemann hypothesis holds (some sources incorrectly claim that Weinberger's results imply that there's at most one other idoneal number).[4]

See also

Notes

  1. ^ Eric Rains, OEISA000926 Comments on A000926, December 2007.
  2. ^ Roberts, Joe: The Lure of the Integers. The Mathematical Association of America, 1992
  3. ^ Acta Arith., 22 (1973), p. 117-124
  4. ^ Ann. Sci. Math. Québec 35, No 2, (2011), 197-227

References