Heegner number

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

Importance of Heegner numbers

In the Gaussian numbers and the Eisenstein numbers, the factorization is essentially unique. One can now ask what other extensions of the integers, this is also the case. Restricting thereby wholeness rings of extensions of the rational numbers by adjoining the square root of a square-free negative integer a, 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, respectively.

Also, the fact that

Has for only prime numbers as values ​​, follows directly from the decomposition law for quadratic number fields, as has class number.

History of the problem

The solution of the problem has already been suggested by Carl Friedrich Gauss. It was known before 1952, that there can be such numbers more than ten. Kurt Heegner finally found that the nine above-mentioned figures are actually all solutions.

• Integer amount
• Number Theory
• Algebraic Number Theory