A note on class-number one in certain real quadratic and pure cubic fields

Let p be any odd prime and let h ( p ) be the class number of the real quadratic field 2(f~p). The results of a computer run to determine the density of the field £(fp) with h(p) = 1 and p < 108 are presented. Similar results are given for pure cubic fields 2(fp) with p < 106. 1. Introduction. Let p be any odd prime and let h = h(p) be the class number of the quadratic field 2(\(p). It is well known that h(p) is odd, but the problem of how frequently h(p) = 1, although it goes back to Gauss, is still unsolved. If we let it (a, b; x) denote the number of primes of the form a + bk less than or equal to x and f(a,b;x) denote the number of these primes p for which h(p) = 1, we find (see Lakein (5)) from a large table of Kuroda (4), that r(l,4;x) =/(l,4;x)/77(l,4;x) = .7765 for x = 2776817; that is, over 77% of all the primes (= 1 (mod4)) up to 2776817 have h(p) = 1. Indeed, according to the recent heuristic results of Cohen and Lenstra (see Cohen (1)), we would expect that h(p) = 1 with probability .75446. In order to test this heuristic, we developed and ran a computer program which determined whether or not h(p) = 1 for all primes p < 108. In the next section of this note we give the results of this computer run. In the following section we present some data for certain pure cubic fields ^(-/p) with p < 106. 2. The Quadratic Case. In order to find h(p), we made use of the well-known formula