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%I M0827 #218 Apr 17 2023 07:27:58
%S 1,2,3,7,11,19,43,67,163
%N Heegner numbers: imaginary quadratic fields with unique factorization (or class number 1).
%C Could also be called Gauss numbers, since he discovered them. Heegner proved list is complete. - _Artur Jasinski_, Mar 21 2003
%C Numbers n such that Q(sqrt(-n)) has unique factorization into primes.
%C These are the squarefree values of n for which if some positive integer N can be written in the form (a/2)^2+n*(b/2)^2 for integers a and b, then every prime factor P of N which occurs to an odd power can also be written in the form (c/2)^2+n*(d/2)^2 for integers c and d. - _V. Raman_, Sep 17 2012, May 01 2013
%C For n = 1 and n = 2, the rings Z[i] (Gaussian Integers), and Z(sqrt(-2)) = numbers of the form a + b*sqrt(-2), where a and b are integers, admit unique factorization. - _V. Raman_, Sep 17 2012
%C For the values of n congruent to 3 (mod 4), the set of numbers of the form (a/2) + (b/2)*sqrt(-n), for integers a and b of the same parity, admit unique factorization. - _V. Raman_, Sep 17 2012, corrected by _Eric M. Schmidt_, Feb 17 2013
%C The Heegner numbers greater than 3 can also be found using the Kronecker symbol, as follows: A number k > 3 is a Heeger number if and only if s = Sum_{j = 1..k} j * (j|k) is prime, which happens to be negative, where (x|y) is the Kronecker symbol. Also note for these results s = -k. But if s = -k is used as the selection condition (instead of primality), then the cubes of {7, 11, 19, 43, 67, 163} are also selected, followed by these same numbers to 9th power (and presumably followed by the 27th or 81st power). - _Richard R. Forberg_, Jul 18 2016
%C Theorem: The ring of integers of the imaginary quadratic field Q(sqrt(-n)) is Euclidean iff n = 1, 2, 3, 7 and 11. (Otherwise, the ring of integers of the imaginary quadratic field Q(sqrt(-n)) is principal iff n is a term of this sequence) [Link Stark-Heegner theorem]. - _Bernard Schott_, Feb 07 2020
%C Named after the German high school teacher and radio engineer Kurt Heegner (1893-1965). - _Amiram Eldar_, Jun 15 2021
%D John H. Conway and Richard K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 224.
%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, p. 213.
%D Wilfred W. J. Hulsbergen, Conjectures in Arithmetic Algebraic Geometry, Vieweg, 1994, p. 8.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D Harold M. Stark, An Introduction to Number Theory. Markham, Chicago, 1970, p. 295.
%H Aram Bingham, <a href="https://arxiv.org/abs/2002.02059">Ternary arithmetic, factorization, and the class number one problem</a>, arXiv:2002.02059 [math.NT], 2020. See p. 9.
%H Kalyan Chakraborty, Azizul Hoque and Richa Sharma, <a href="https://arxiv.org/abs/1812.11874">Complete solutions of certain Lebesgue-Ramanujan-Nagell type equations</a>, arXiv:1812.11874 [math.NT], 2018.
%H Alex Clark and Brady Haran, <a href="https://www.youtube.com/watch?v=DRxAVA6gYMM">163 and Ramanujan Constant</a>, Numberphile video, 2012.
%H Noam Elkies, <a href="https://dash.harvard.edu/handle/1/2920120">The Klein quartic in number theory</a>, in: S. Levy, ed., The Eightfold Way, Cambridge Univ. Press, 1999, pp. 51-101. MR1722413 (2001a:11103). See page 93.
%H Yang-Hui He and John McKay, <a href="http://arxiv.org/abs/1505.06742">Sporadic and Exceptional</a>, arXiv:1505.06742 [math.AG], 2015.
%H Kurt Heegner, <a href="http://dx.doi.org/10.1007/BF01174749">Diophantische Analysis und Modulfunktionen</a>, Matematische Zaitschrift, Vol. 56 (1952), pp. 227-253.
%H John Myron Masley, <a href="http://dx.doi.org/10.1007/BFb0062711">Where are the number fields with small class number?</a>, in: M. B. Nathanson (ed.), Number Theory Carbondale 1979, Lect. Notes Math., Vol. 751, Springer, Berlin, Heidelberg, 1982, pp. 221-242.
%H Rick L. Shepherd, <a href="http://libres.uncg.edu/ir/uncg/f/Shepherd_uncg_0154M_11099.pdf">Binary quadratic forms and genus theory</a>, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GausssClassNumberProblem.html">Gauss's Class Number Problem</a> and <a href="http://mathworld.wolfram.com/HeegnerNumber.html">Heegner Number</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Heegner_number">Heegner number</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Stark%E2%80%93Heegner_theorem">Stark-Heegner theorem</a>.
%H <a href="/index/Qua#quadfield">Index entries for sequences related to quadratic fields</a>
%F a(n) = A263465(n) = -A048981(6-n) for n <= 5. - _Jonathan Sondow_, May 28 2016
%t Union[ Select[ -NumberFieldDiscriminant[ Sqrt[-#]]& /@ Range[200], NumberFieldClassNumber[ Sqrt[-#]] == 1 & ] /. {4 -> 1, 8 -> 2}] (* _Jean-François Alcover_, Jan 04 2012 *)
%t heegnerNums = {}; Do[s = Sum[j * KroneckerSymbol[j, k], {j, 1, k}]; If[PrimeQ[s], AppendTo[heegnerNums, {s, k}]], {k, 1, 10000}]; heegnerNums (* _Richard R. Forberg_, Jul 18 2016 *)
%o (PARI) select(n->qfbclassno(-n*if(n%4==3,1,4))==1, vector(200,i,i)) \\ _Charles R Greathouse IV_, Nov 20 2012
%Y Cf. A003174, A005847 (for class number 2), A014602 (for discriminants of these fields), A048981, A263465.
%K nonn,fini,full,nice
%O 1,2
%A _N. J. A. Sloane_
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