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A000926
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Euler's "numerus idoneus" (or "numeri idonei", or idoneal, or suitable, or convenient numbers).
(Formerly M0476 N0176)
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52
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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
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OFFSET
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1,2
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COMMENTS
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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 are 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
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REFERENCES
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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.
P. Ribenboim, 'My Numbers, My Friends', Chap.11 Springer-Verlag 2000 NY
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).
A. Weil, Number theory: an approach through history; from Hammurapi to Legendre, Birkhäuser, Boston, 1984; see pp. 188, 219-226.
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LINKS
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David Masser, Alan Baker, arXiv:2010.10256 [math.HO], 2020. See p. 24.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A139642 (congruences for idoneal quadratic forms).
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KEYWORD
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nonn,fini,full,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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