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Idoneal number

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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, prime power, twice one of these, or a power of 2. 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 integer a, b, and c.[1]

It is sufficient to consider the set { n + k2 | k2 ≤ 3 · 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:
Is there a 66th idoneal number?
(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).

In 1973, Peter J. Weinberger proved that at most one other idoneal number exists, and that the list above is complete if the generalized Riemann hypothesis holds.[3]

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

References

  • Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966, pp. 425–430.
  • D. Cox, "Primes of Form x2 + n y2", Wiley, 1989, p. 61.
  • 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.
  • K. S. Brown, Mathpages, Numeri Idonei
  • M. Waldschmidt, Open Diophantine problems
  • Weisstein, Eric W. "Idoneal Number". MathWorld.
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