Rabinowitsch Revisited

In the late eighteenth century both Euler and Legendre noticed that n +n+41 is prime for n = 0, 1, 2 . . . 39, and remarked that there are few polynomials with such small degree and coefficients that give such a long string of consecutive prime values. Rabinowitsch, at the 1912 International Congress of Mathematicians [18], showed that n + n + A is prime for n = 0, 1, 2, . . . A − 2 if and only if 4A − 1 is squarefree and the ring of integers of the field Q( √ 1 − 4A) has just one equivalence class of ideals (that is, class number one). In 1934 Heilbronn [11] proved that there are only finitely many such fields, and in 1952 Heegner [10] that there are just seven such fields, corresponding to A = 1, 2, 3, 5, 11, 17 and 41. One can generalize Rabinowitsch’s criterion to other polynomials, and to other fields; for example, Mollin and Williams proved the following for real quadratic fields: n + n − A is prime for all positive n < √ A − 1 if and only if the field Q( √ 4A + 1) has class number one where either A = 4, or A ≥ 5 is odd and is of the form m or m + m± 1 for some integer m, see [17, 352-354]. One can develop similar criterion for when the class number is 2, or 3, or any fixed number (see [15, 16]). The idea in all of these proofs is that if a large proportion of the values of a quadratic polynomial of discriminant d are prime then there cannot be many small primes p for which (d/p) = 1 (else those small primes would divide the values of the given quadratic polynomial, preventing it from being prime very often). If that is the case then the value of L(1, (d/.)) will be surprisingly small, which is equivalent to having h(d), the class number, small if d < 0, and to having both h(d) and d, the fundamental unit, small if d > 0. We remark that d is “small” if and only if the continued fraction for (1 + √ d)/2 or √ d/2 (as d ≡ 1 or 0 (mod 4)) is short, that is if d is a value of one of several special forms. Siegel [22] showed that L(1, (d/.)) 1/d and Tatuzawa [23] made Siegel’s argument explicit, excluding at most one d, a presumably hypothetical counterexample to the Generalized Riemann Hypothesis. Using Tatuzawa’s result, Mollin [15, 16] gives many explicit criteria “with one possible exception”. One might ask whether it is possible to find quadratic polynomials with arbitrarily long strings of consecutive prime values (though we do not necessarily constrain