Idoneal Number

An idoneal number, also called a suitable number or convenient number, is a positive integer  for which the fact that a number is a monomorph (i.e., is expressible in only one way as  where  is relatively prime to ) 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  is idoneal iff it cannot be written as  for integer , , and  with .

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.

REFERENCES:

Borevich, Z. I. and Shafarevich, I. R. Number Theory. New York: Academic Press, pp. 425-430, 1966.

Cox, D. "Primes of Form ." 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.

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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.

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Steinig, J. "On Euler's Idoneal Numbers." Elemente Math. 21, 73-88, 1966.

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