Search: id:A000926
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%I A000926 M0476 N0176
%S A000926 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,
%T A000926 45,48,57,58,60,70,72,78,85,88,93,102,105,112,120,130,133,165,168,177,
%U A000926 190,210,232,240,253,273,280,312,330,345,357,385,408,462,520,760,840,1320,
               1365,1848
%N A000926 Euler's "numerus idoneus" (idoneal, or suitable, or convenient numbers).
%C A000926 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:
%C A000926 (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]
%C A000926 (2) Every genus of quadratic forms of discriminant -4n consists of a 
               single class. [Gauss]
%C A000926 (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]
%C A000926 (4) Two quadratic forms of discriminant -4n are equivalent if and only 
               if they are properly equivalent. [Cox]
%C A000926 (5) The class group C(-4n) is isomorphic to (Z/2Z)^m for some integer 
               m. [Cox]
%C A000926 (6) n is not of the form ab+ac+bc with 0 < a < b < c. [Rains]
%C A000926 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.
%C A000926 If an additional term exists it is > 100000000. - Jud McCranie (j.mccranie(AT)comcast.net), 
               Jun 27 2005
%C A000926 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.
%C A000926 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 (noe(AT)sspectra.com), 
               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]
%D A000926 Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, 
               NY, 1966, pp. 425-430.
%D A000926 S. Chowla, An extension of Heilbronn's class number theorem, Quart. J. 
               math., 5 (1934), 304-307.
%D A000926 D. Cox, "Primes of Form x^2 + n y^2", Wiley, 1989, Section 3.
%D A000926 J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 1848, p. 146, 
               Ellipses, Paris 2008.
%D A000926 G. Frei, Euler's convenient numbers, Math. Intell. Vol. 7 No. 3 (1985), 
               55-58 and 64.
%D A000926 C. F. Gauss, Disquisitiones Arithmeticae, 1801. English translation: 
               Yale University Press, New Haven, CT, 1966, Sections 329-334.
%D A000926 O.-H. Keller, Ueber die "Numeri idonei" von Euler, Beitraege Algebra 
               Geom., 16 (1983), 79-91. [Math. Rev. 85m:11019]
%D A000926 G. B. Mathews, Theory of Numbers, Chelsea, no date, p. 263.
%D A000926 P. Ribenboim, "Galimatias Arithmeticae", in Mathematics Magazine 71(5) 
               339 1998 MAA or, 'My Numbers, My Friends', Chap.11 Springer-Verlag 
               2000 NY
%D A000926 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000926 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000926 J. Steinig, On Euler's ideoneal numbers, Elemente Math., 21 (1966), 73-88.
%D A000926 A. Weil, Number theory: an approach through history; from Hammurapi to 
               Legendre, Birkhaeuser, Boston, 1984; see pp. 188, 219-226.
%D A000926 P. Weinberger, Exponents of the class groups of complex quadratic fields, 
               Acta Arith., 22 (1973), 117-124.
%H A000926 K. S. Brown, Mathpages, <a href="http://www.mathpages.com/home/kmath058.htm">
               Numeri Idonei</a>
%H A000926 Eric Rains, <a href="a000926.txt">Comments on A000926</a>
%H A000926 M. Waldschmidt, <a href="http://arXiv.org/abs/math.NT/0312440">Open Diophantine 
               problems</a>
%H A000926 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               IdonealNumber.html">Idoneal Number</a>
%t A000926 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 (noe(AT)sspectra.com), Apr 08 2004
%o A000926 (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
%Y A000926 Sequence A025052 is a subsequence.
%Y A000926 Cf. A014556, A026501, A093669, A094376, A094377, A094378.
%Y A000926 Cf. A139642 (congruences for idoneal quadratic forms).
%Y A000926 Sequence in context: A049812 A093668 A026501 this_sequence A011875 A053433 
               A091401
%Y A000926 Adjacent sequences: A000923 A000924 A000925 this_sequence A000927 A000928 
               A000929
%K A000926 nonn,fini,nice
%O A000926 1,2
%A A000926 N. J. A. Sloane (njas(AT)research.att.com).
%E A000926 Edited by N. J. A. Sloane (njas(AT)research.att.com), Dec 07 2007

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