He is famous for his mathematical discoveries in number theory and, in particular, the Stark–Heegner theorem.
In 1952, he published the Stark–Heegner theorem which he claimed was the solution to a classic number theory problem proposed by the great mathematician Gauss, the class number 1 problem.
Heegner's proof was accepted as essentially correct after a 1967 announcement by Bryan Birch, and definitively resolved by a paper by Harold Stark that had been delayed in publication until 1969 (Stark had independently arrived at a similar proof, but disagrees with the common notion that his proof is "more or less the same" as Heegner's).
[1] Stark attributed Heegner's mistakes to the fact he used a textbook by Weber that contained some results with incomplete proofs.
The book The Legacy of Leonhard Euler: A Tricentennial Tribute by Lokenath Debnath claims on page 64, that Heegner was a "retired Swiss mathematician", but he appears to have been neither Swiss nor retired at the time of his 1952 paper.[2][3][relevant?]