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. 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.

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]

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

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.

See also

List of unsolved problems in mathematics

Notes

Eric Rains, OEIS A000926 Comments on A000926, December 2007.

Roberts, Joe: The Lure of the Integers. The Mathematical Association of America, 1992

References

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966, pp. 425–430.
D. Cox, "Primes of Form x² + n y²", Wiley, 1989, p. 61.
L. Euler, "An illustration of a paradox about the idoneal, or suitable, numbers", 1806
G. Frei, Euler's convenient numbers, Math. Intell. Vol. 7 No. 3 (1985), 55–58 and 64.
O-H. Keller, Ueber die "Numeri idonei" von Euler, Beitraege Algebra Geom., 16 (1983), 79–91. [Math. Rev. 85m:11019]
G. B. Mathews, Theory of Numbers, Chelsea, no date, p. 263.
P. Ribenboim, "Galimatias Arithmeticae", in Mathematics Magazine 71(5) 339 1998 MAA or, 'My Numbers, My Friends', Chap.11 Springer-Verlag 2000 NY
J. Steinig, On Euler's ideoneal numbers, Elemente Math., 21 (1966), 73–88.
A. Weil, Number theory: an approach through history; from Hammurapi to Legendre, Birkhaeuser, Boston, 1984; see p. 188.
P. Weinberger, Exponents of the class groups of complex quadratic fields, Acta Arith., 22 (1973), 117–124.

External links

K. S. Brown, Mathpages, Numeri Idonei
M. Waldschmidt, Open Diophantine problems
Weisstein, Eric W., "Idoneal Number", MathWorld.

Mathematics Encyclopedia