Idoneal Number

A Positive value of $D$ for which the fact that a number is a Monomorph (i.e., the number is expressible in only one way as $x^2+Dy^2$ or $x^2-Dy^2$ where $x^2$ is Relatively Prime to $Dy^2$) guarantees it to be a Prime, Power of a Prime, or twice one of these. The numbers are also called Euler's Idoneal Numbers, or Suitable 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 (Sloane's A000926).

References

Shanks, D. ``On Gauss's Class Number Problems.'' Math. Comput. 23, 151-163, 1969.

Sloane, N. J. A. Sequence A000926/M0476 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.