

A003173


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


56




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 * (jk) is prime, which happens to be negative, where (xy) 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 StarkHeegner theorem].  Bernard Schott, Feb 07 2020
Named after the German high school teacher and radio engineer Kurt Heegner (18931965).  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

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. 221242.


FORMULA



MATHEMATICA

Union[ Select[ NumberFieldDiscriminant[ Sqrt[#]]& /@ Range[200], NumberFieldClassNumber[ Sqrt[#]] == 1 & ] /. {4 > 1, 8 > 2}] (* JeanFrançois 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



CROSSREFS



KEYWORD

nonn,fini,full,nice


AUTHOR



STATUS

approved



