A003173 Heegner numbers: imaginary quadratic fields with unique factorization (or class number 1). (Formerly M0827)

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

Offset

1,2

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

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

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

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

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.

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

Table of n, a(n) for n=1..9.

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

Formula

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A158709 A180422 A055502 * A159262 A160434 A139630

Adjacent sequences: A003170 A003171 A003172 * A003174 A003175 A003176

Keyword

nonn,fini,full,nice

Author

N. J. A. Sloane

Status

approved