Idoneal number

In mathematics, Euler's idoneal numbers (also called suitable numbers or convenient numbers) are the positive integers D such that any integer expressible in only one way as x2 ± Dy2 (where x2 is relatively prime to Dy2) is a prime power or twice a prime power. In particular, a number that has two distinct representations as a sum of two squares is composite. Every idoneal number generates a set containing infinitely many primes and missing infinitely many other primes.

Definition

A positive integer n is idoneal if and only if it cannot be written as ab + bc + ac for distinct positive integers a, b, and c.[1]

It is sufficient to consider the set { n + k2 | 3 . k2 ≤ n ∧ gcd (n, k) = 1 }; if all these numbers are of the form p, p2, 2 · p or 2s for some integer s, where p is a prime, then n is idoneal.[2]

Conjecturally complete listing
Unsolved problem in mathematics:

Are there 65, 66 or 67 idoneal numbers?

(more unsolved problems in mathematics)

The 65 idoneal numbers found by Leonhard Euler and Carl Friedrich Gauss and conjectured to be the only such numbers 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 (sequence A000926 in the OEIS).

Results of Peter J. Weinberger from 1973[3] imply that at most two other idoneal numbers exist, and that the list above is complete if the generalized Riemann hypothesis holds (some sources incorrectly claim that Weinberger's results imply that there is at most one other idoneal number).[4]

See also

Notes
  1. Eric Rains, OEIS: A000926 Comments on A000926, December 2007.
  2. Roberts, Joe: The Lure of the Integers. The Mathematical Association of America, 1992
  3. Acta Arith., 22 (1973), p. 117-124
  4. Kani, Ernst (2011). "Idoneal numbers and some generalizations" (PDF). Annales des Sciences Mathématiques du Québec. 35 (2). Corollary 23, Remark 24.

References
  • Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966, pp. 425–430.
  • D. A. Cox (1989). Primes of the Form x2 + ny2. Wiley-Interscience. p. 61. ISBN 0-471-50654-0.
  • L. Euler, "An illustration of a paradox about the idoneal, or suitable, numbers", 1806
  • G. Frei, Euler's convenient numbers, Math. Intell. Vol. 7 No. 3 (1985), 55–58 and 64.
  • O-H. Keller, Ueber die "Numeri idonei" von Euler, Beitraege Algebra Geom., 16 (1983), 79–91. [Math. Rev. 85m:11019]
  • G. B. Mathews, Theory of Numbers, Chelsea, no date, p. 263.
  • P. Ribenboim, "Galimatias Arithmeticae", in Mathematics Magazine 71(5) 339 1998 MAA or, 'My Numbers, My Friends', Chap.11 Springer-Verlag 2000 NY
  • J. Steinig, On Euler's ideoneal numbers, Elemente Math., 21 (1966), 73–88.
  • A. Weil, Number theory: an approach through history; from Hammurapi to Legendre, Birkhaeuser, Boston, 1984; see p. 188.
  • P. Weinberger, Exponents of the class groups of complex quadratic fields, Acta Arith., 22 (1973), 117–124.
  • Ernst Kani, Idoneal Numbers And Some Generalizations, Ann. Sci. Math. Québec 35, No 2, (2011), 197-227.
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