An idoneal number, also called a suitable number or convenient number, is a positive integer
with the following property. If an odd integer is relatively prime
to , the equation has exactly one solution in nonnegative
integers
and ,
and
and
are relatively prime, then is a prime number (Kani 2011,
p. 198). These numbers are also called Euler's idoneal numbers.
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 are at most two more.
Under the generalized Riemann hypothesis,
there are no additional idoneal numbers (Kani 2011, pp. 208-209).
Borevich, Z. I. and Shafarevich, I. R. Number Theory. New York: Academic Press, pp. 425-430, 1966.Cox,
D. A. Primes
of the Form x2+ny2: Fermat, Class Field
Theory and Complex Multiplication. New York: Wiley, p. 61, 1989.Frei,
G. "Euler's Convenient Numbers." Math. Intell.7, 55-58 and
64, 1985.Kani, E. "Idoneal Numbers and Some Generalizations."
Ann. Sci. Math. Québec35, 197-227, 2011. https://www.labmath.uqam.ca/~annales/volumes/35-2/PDF/197-227.pdf.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,
MA: Birkhäuser, p. 188, 1984.Weinberger, P. J. "Exponents
of the Class Groups of Complex Quadratic Fields." Acta Arith.22,
117-124, 1973.
with the following property. If an odd integer 
is relatively prime
to 
, the equation 
has exactly one solution in nonnegative
integers 
and 
,
and 
and 
are relatively prime, then 
is a prime number (Kani 2011,
p. 198). These numbers are also called Euler's idoneal numbers.
is idoneal iff it cannot be written
as 
for integers 
, 
,
and 
with 
.
