Heegner number

The Heegner numbers are the nine numbers 1, 2, 3, 7, 11, 19, 43, 67 and 163. They are named after Kurt Heegner .

Meaning of the Heegner numbers

In the Gaussian numbers and in the Eisenstein numbers , the prime factorization is essentially unique. One can now ask for which other expansions of the integers this is also the case. If one restricts oneself to wholeness rings of extensions of the rational numbers through the adjunction of the square root of a square-free negative integer , it turns out that the prime factorization is unique if and only if a Heegner number is. The Gaussian numbers and the Eisenstein numbers correspond to the cases and . ${\displaystyle \mathbb {Q} ({\sqrt {d}})}$ ${\displaystyle d}$ ${\displaystyle -d}$ ${\displaystyle d=-1}$ ${\displaystyle d=-3}$

Also the fact that

${\displaystyle N\left({\frac {2X\pm 1+{\sqrt {-163}}}{2}}\right)=\left({\frac {2X\pm 1}{2}}\right)^{2}+\left({\frac {\sqrt {163}}{2}}\right)^{2}=X^{2}\pm X+{\frac {1+163}{4}}=X^{2}\pm X+41}$

for only has prime numbers as values, follows directly from the decomposition law for quadratic number fields , since has class number . ${\displaystyle X=0,1,\ldots ,{\frac {1+163}{4}}-1=40}$ ${\displaystyle \mathbb {Q} \left({\sqrt {-163}}\right)}$ ${\displaystyle 1}$

History of the problem

The solution to the problem has already been suggested by Carl Friedrich Gauß . It was known before 1952 that there could be a maximum of ten such numbers. Kurt Heegner finally found that the nine numbers mentioned above are actually all the solutions.

Other references

• Generate Heegner numbers Fast integers (Almost Integer) , z. B. the Ramanujan constant . ${\displaystyle e^{\pi {\sqrt {163}}}}$
• The Heegner numbers are linked to the j function and generate cube numbers using these.

Web links

• Eric W. Weisstein : Heegner number . In: MathWorld (English).
• Follow A003173 in OEIS with further references
• Dorian Goldfeld : Gauss' Class Number Problem for Imaginary Quadratic Fields. (detailed history in the Bulletin of the American Mathematical Society, 1985) (PDF; 1.1 MB, 16 pages)

Individual evidence

1. ↑ Eric W. Weisstein : Ramanujan Constant . In: MathWorld (English). see. also en: Almost integer
2. ↑ Eric W. Weisstein : j-Function . In: MathWorld (English).