Idoneal number

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, prime power, or twice one of these.

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

The 65 idoneal numbers found by Carl Friedrich Gauss and Leonhard Euler 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 OEIS). Weinberger proved in 1973 that at most one other idoneal number exists, and that if the generalized Riemann hypothesis holds, then the list is complete.

References

  1. ^ Eric Rains, Comments on A000926, December 2007.

External links