A003173 Heegner numbers: imaginary quadratic fields with unique factorization (or class number 1).

1, 2, 3, 7, 11, 19, 43, 67, 163

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• fini
• full
• nice

Comments

Could also be called Gauss numbers, since he discovered them. Heegner proved list is complete. - Artur Jasinski, Mar 21 2003

Numbers n such that Q(sqrt(-n)) has unique factorization into primes.

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

Cases n = 1 and n = 2 correspond to the rings Z[i] (Gaussian integers) and Z[sqrt(-2)] = numbers of the form a + b*sqrt(-2), where a and b are integers. Other cases, satisfying a(n) == 3 (mod 4), correspond to the rings of numbers of the form (a/2) + (b/2)*sqrt(-a(n)), for integers a and b of the same parity. All these rings admit unique factorization. - V. Raman, Sep 17 2012, corrected by Eric M. Schmidt, Feb 17 2013

The Heegner numbers greater than 3 can also be found using the Kronecker symbol, as follows: A number k > 3 is a Heegner 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

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

Named after the German high school teacher and radio engineer Kurt Heegner (1893-1965). - Amiram Eldar, Jun 15 2021

References

• John H. Conway and Richard K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 224.
• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, p. 213.
• Wilfred W. J. Hulsbergen, Conjectures in Arithmetic Algebraic Geometry, Vieweg, 1994, p. 8.
• Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 143.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• Harold M. Stark, An Introduction to Number Theory. Markham, Chicago, 1970, p. 295.

Links

• Aram Bingham, Ternary arithmetic, factorization, and the class number one problem, arXiv:2002.02059 [math.NT], 2020. See p. 9.
• Kalyan Chakraborty, Azizul Hoque and Richa Sharma, Complete solutions of certain Lebesgue-Ramanujan-Nagell type equations, arXiv:1812.11874 [math.NT], 2018.
• Alex Clark and Brady Haran, 163 and Ramanujan Constant, Numberphile video, 2012.
• Noam Elkies, The Klein quartic in number theory, in: S. Levy, ed., The Eightfold Way, Cambridge Univ. Press, 1999, pp. 51-101. MR1722413 (2001a:11103). See page 93.
• Yang-Hui He and John McKay, Sporadic and Exceptional, arXiv:1505.06742 [math.AG], 2015.
• Kurt Heegner, Diophantische Analysis und Modulfunktionen, Matematische Zaitschrift, Vol. 56 (1952), pp. 227-253.
• John Myron Masley, Where are the number fields with small class number?, in: M. B. Nathanson (ed.), Number Theory Carbondale 1979, Lect. Notes Math., Vol. 751, Springer, Berlin, Heidelberg, 1982, pp. 221-242.
• Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.
• Eric Weisstein's World of Mathematics, Gauss's Class Number Problem and Heegner Number.
• Wikipedia, Heegner number.
• Wikipedia, Stark-Heegner theorem.
• Index entries for sequences related to quadratic fields

Crossrefs

Cf. A003174, A005847 (for class number 2), A014602 (for discriminants of these fields), A048981, A263465.

Programs

Mathematica

Union[ Select[ -NumberFieldDiscriminant[ Sqrt[-#]]& /@ Range[200], NumberFieldClassNumber[ Sqrt[-#]] == 1 & ] /. {4 -> 1, 8 -> 2}] (* Jean-François Alcover, Jan 04 2012 *)
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 *)

PARI

select(n->qfbclassno(-n*if(n%4==3,1,4))==1, vector(200,i,i)) \\ Charles R Greathouse IV, Nov 20 2012

Formula

a(n) = A263465(n) = -A048981(6-n) for n <= 5. - Jonathan Sondow, May 28 2016