In number theory, a Heegner number (as termed by Conway and Guy) is a square-free positive integer d such that the imaginary quadratic field has class number 1. Equivalently, the ring of algebraic integers of has unique factorization.[1]
The determination of such numbers is a special case of the class number problem, and they underlie several striking results in number theory.
According to the (Baker–)Stark–Heegner theorem there are precisely nine Heegner numbers:
1, 2, 3, 7, 11, 19, 43, 67, and 163. (sequence A003173 in the OEIS)
This result was conjectured by Gauss and proved up to minor flaws by Kurt Heegner in 1952. Alan Baker and Harold Stark independently proved the result in 1966, and Stark further indicated the gap in Heegner's proof was minor.[2]
^Conway, John Horton; Guy, Richard K. (1996). The Book of Numbers. Springer. p. 224. ISBN 0-387-97993-X.
^Stark, H. M. (1969), "On the gap in the theorem of Heegner" (PDF), Journal of Number Theory, 1 (1): 16–27, Bibcode:1969JNT.....1...16S, doi:10.1016/0022-314X(69)90023-7, hdl:2027.42/33039
In number theory, a Heegnernumber (as termed by Conway and Guy) is a square-free positive integer d such that the imaginary quadratic field Q [ − d ]...
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trillion digits on March 14, 2024. The algorithm is based on the negated Heegnernumber d = − 163 {\displaystyle d=-163} , the j-function j ( 1 + i 163 2 )...
which is also implicitly quadratic, and the class number; this polynomial is related to the Heegnernumber 163 = 4 ⋅ 41 − 1 {\displaystyle 163=4\cdot 41-1}...
explanation for this phenomenon led to the deep algebraic number theory of Heegner numbers and the class number problem. The Hardy–Littlewood conjecture F predicts...
{\displaystyle \mathbb {Q} [{\sqrt {-d}}]} will fail to be a UFD unless d is a Heegnernumber. The ring of formal power series over the complex numbers is a UFD,...
for most others here). It is a consequence of the fact that 163 is a Heegnernumber. There are several integers k = 2198 , 422151 , 614552 , 2508952 , 6635624...
congruent to 2 modulo 3, but no lucky numbers are congruent to 2 modulo 3. Heegnernumber List of topics named after Leonhard Euler Formula for primes Ulam spiral...
and proven by Kurt Heegner, although Heegner's proof was not believed until Harold Stark gave a later proof in 1967. (See Stark–Heegner theorem.) This is...
L-series of an elliptic curve evaluated at 1 to the height of a certain Heegner point. This theorem has some applications, including implying cases of...
formulated ideas on the role of Heegner points (he was one of those reconsidering Kurt Heegner's original work on the class number one problem, which had not...
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