Class numbers of quadratic fields If is a square-free integer (a product of distinct primes) other than 1, then is a quadratic extension of Q. If , then the class number of the ring of algebraic integers of is equal to 1 for precisely the following values of : . This result was first conjectured by Gauss 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 a special case of the famous class number problem.
clAss numBErs oF QUADRaTic FiElds if Is a sQuARe-free intEgER (a prODuct Of disTiNcT prImes) oTHER ThaN 1, tHEN is a quadratiC exTeNsion Of Q. if , then the cLass nuMbEr oF thE rinG of AlGebraIc integers of iS equaL tO 1 for pREcisEly THe FollOwinG vaLueS Of : . THiS resuLt wAS fIrst cOnjeCtured by gauss 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 A special case of tHe fAmous claSS Number ProbLeM.