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Idoneal Number


An idoneal number, also called a suitable number or convenient number, is a positive integer D for which the fact that a number is a monomorph (i.e., is expressible in only one way as x^2+/-Dy^2 where x^2 is relatively prime to Dy^2) guarantees it to be a prime, prime power, or twice one of these. The 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+ca for integer a, b, and c with 0<a<b<c.

The 65 idoneal numbers found by Gauss and Euler and conjectured to be the only such numbers (Shanks 1969) 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 (OEIS A000926). It is known that if any other idoneal numbers exist, there can be only one more.


See also

Monomorph

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References

Borevich, Z. I. and Shafarevich, I. R. Number Theory. New York: Academic Press, pp. 425-430, 1966.Cox, D. "Primes of Form x^2+ny^2." New York: Wiley, p. 61, 1989.Frei, G. "Euler's Convenient Numbers." Math. Intell. 7, 55-58 and 64, 1985.Keller, O.-H. "Über die 'Numeri idonei' von Euler." Beiträge Algebra Geom. 16, 79-91, 1983.Mathews, G, B. Theory of Numbers. Chelsea, p. 263.Ribenboim, P. "Galimatias Arithmeticae." Math. Mag. 71, 339, 1998.Ribenboim, P. Ch. 11 in My Numbers, My Friends. New York: Springer-Verlag, 2000.Shanks, D. "On Gauss's Class Number Problems." Math. Comput. 23, 151-163, 1969.Sloane, N. J. A. Sequence A000926/M0476 in "The On-Line Encyclopedia of Integer Sequences."Steinig, J. "On Euler's Idoneal Numbers." Elemente Math. 21, 73-88, 1966.Weil, A. Number Theory: an Approach Through History, from Hammurapi to Legendre. Boston: Birkhäuser, p. 188, 1984.Weinberger, P. "Exponents of the Class Groups of Complex Quadratic Fields.' Acta Arith. 22, 117-124, 1973.

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Idoneal Number

Cite this as:

Weisstein, Eric W. "Idoneal Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/IdonealNumber.html

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