A000926 Euler's "numerus idoneus" (or "numeri idonei", or idoneal, or suitable, or convenient numbers).

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, 1848

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• fini
• full
• nice

Comments

There are many equivalent definitions of these numbers. Based on Cox, Theorem 3.22 and Proposition 3.24 and a comment by Eric Rains (rains(AT)caltech.edu), we can say that a positive number n belongs to this sequence if and only if any of the following equivalent statements is true:

(1) Let m > 1 be an odd number relatively prime to n which can be written in the form x^2 + n*y^2 with x, y relatively prime. If the equation m = x^2 + n*y^2 has only one solution with x, y >= 0, then m is a prime number. [Euler]

(2) Every genus of quadratic forms of discriminant -4n consists of a single class. [Gauss]

(3) If a*x^2 + b*x*y + c*y^2 is a reduced quadratic form of discriminant -4n, then either b=0, a=b or a = c. [Cox]

(4) Two quadratic forms of discriminant -4n are equivalent if and only if they are properly equivalent. [Cox]

(5) The class group C(-4n) is isomorphic to (Z/2Z)^m for some integer m. [Cox]

(6) n is not of the form ab+ac+bc with 0 < a < b < c. (See proof in link below.) [Rains]

It is conjectured that the list given here is complete. Chowla showed that the list is finite and Weinberger showed that there is at most one further term.

If an additional term exists it is > 100000000. - Jud McCranie, Jun 27 2005

The terms shown are the union of {1,2,3,4,7}, A033266, A033267, A033268 and A033269 (corresponding to class numbers 1, 2, 4, 8 and 16 respectively).

Note that for n in this sequence, n+1 is either a prime, twice a prime, the square of a prime, 8 or 16. - T. D. Noe, Apr 08 2004. [This is a general theorem that is not hard to prove using genus theory. The "32" in the original comment was an error. - Tom Hagedorn (hagedorn(AT)tcnj.edu), Dec 29 2008]

Also numbers n such that for all primes p such that p is a quadratic residue (mod 4*n) and p-n is a quadratic residue (mod 4*n), p can be uniquely written into the form as x^2+n*y^2. - V. Raman, Nov 25 2013

References

• Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 97 at p. 272.
• Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966, pp. 425-430.
• David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, Section 3.
• J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 1848, p. 146, Ellipses, Paris 2008.
• C. F. Gauss, Disquisitiones Arithmeticae, 1801. English translation: Yale University Press, New Haven, CT, 1966, Sections 329-334.
• G. B. Mathews, Theory of Numbers, Chelsea, no date, p. 263.
• Paulo Ribenboim, My Numbers, My Friends, Chapter 11, Springer-Verlag, NY, 2000.
• Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 142-143.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 103.
• A. Weil, Number theory: an approach through history; from Hammurapi to Legendre, Birkhäuser, Boston, 1984; see pp. 188, 219-226.

Links

• S. Chowla, An extension of Heilbronn's class number theorem, Quart. J. math., 5 (1934), 304-307.
• K. S. Brown, Mathpages, Numeri Idonei
• Günther Frei, Les nombres convenables de Leonhard Euler, Publications Université de Besançon, 1983-1984.
• Günther Frei, Euler's convenient numbers, Math. Intell. Vol. 7 No. 3 (1985), 55-58 and 64.
• E. Hertel, C. Richter, Tiling Convex Polygons with Congruent Equilateral Triangles, Discrete & Computational Geometry, 2014, DOI 10.1007/s00454-014-9576-7. Mentions this sequence. - _N. J. A. Sloane_, Mar 17 2014
• O.-H. Keller, Über die "Numeri idonei" von Euler, Beitraege Algebra Geom., 16 (1983), 79-91. [Math. Rev. 85m:11019]
• Robert Krzyzanowski, Euler's Convenient Numbers
• David Masser, Alan Baker, arXiv:2010.10256 [math.HO], 2020. See p. 24.
• Eric Rains, Comments on A000926
• Paulo Ribenboim, Galimatias Arithmeticae, in Mathematics Magazine 71(5) 339 1998 MAA.
• Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
• J. Steinig, On Euler's ideoneal numbers, Elemente Math., 21 (1966), 73-88.
• M. Waldschmidt, Open Diophantine problems, arXiv:math/0312440 [math.NT], 2003-2004
• P. Weinberger, Exponents of the class groups of complex quadratic fields, Acta Arith., 22 (1973), 117-124.
• Eric Weisstein's World of Mathematics, Idoneal Number

Crossrefs

Sequence A025052 is a subsequence.

Cf. A014556, A026501, A093669, A094376, A094377, A094378.

Cf. A139642 (congruences for idoneal quadratic forms).

Programs

Mathematica

noSol={}; Do[lim=Ceiling[(n-2)/3]; found=False; Do[If[n>a*b && Mod[n-a*b, a+b]==0 && Quotient[n-a*b, a+b]>b, found=True; Break[]], {a, 1, lim-1}, {b, a+1, lim}]; If[ !found, AppendTo[noSol, n]], {n, 10000}]; noSol (* T. D. Noe, Apr 08 2004 *)

PARI

A000926(Nmax=1e9)={for(n=1,Nmax,for(a=1,sqrtint(n\3),for(b=a+1,(n-a)\(3*a+2),n-a<(2*a+1+b)*b & break;(n-a*b)%(a+b)==0 & next(3)));print1(n", "))} \\ M. F. Hasler, Dec 04 2007

PARI

ok(n)=!#select(k->k<>2, quadclassunit(-4*n).cyc) \\ Andrew Howroyd, Jun 08 2018

Extensions

Edited by N. J. A. Sloane, Dec 07 2007