Idoneal number

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An idoneal number, also called a suitable number or convenient number, is a positive integer D such that any integer expressible in only one way as $x^2\pm Dy^2$ (where x² is relatively prime to Dy²) is a prime, prime power, or twice one of these. These numbers are also called Euler's idoneal numbers or suitable numbers.

A positive integer n is idoneal iff 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. If the generalized Riemann hypothesis holds, then the list is complete.^[2]

[edit] See also

Monomorph

[edit] 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.
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.

^ Eric Rains, Comments on A000926, December 2007.
^ R. A. Mollin, Quadratics, CRC Press, Boca Raton, New York (1995), pp. 172–186; cited in Mollin 1997, Prime-producing quadratics.