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.
."
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.