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Mollin’s conjecture. (English) Zbl 1125.11059
Let \(d\) denote a positive square free integer and \(h(d)\) denote the class number of the quadratic field \(\mathbb Q(\sqrt{d})\). In [Proc. Japan Acad., Ser. A 63, 121–125 (1987; Zbl 0625.12002)], R. A. Mollin conjectured that \(h(n^2- 4)> 1\) if \(n> 21\). The proof combines a new theorem with the well known class number 1 criterion: Let \(d= n^2 - 4> 5\) be square free. Then \(h(d)= 1\) if and only if \(({d\over q})=-1\) for all primes \(q< n- 2\). The new theorem states that if \(h(d)= 1\) for such \(d\) then \(({d\over q})=0\) or \(1\) for at least one \(q= 5,7,61\) or \(1861\). Thus it is immediate that \(h(n^2- 4)> 1\) for \(n> 1863\). The smaller values can be verified either using class number tables or by checking that the two conditions are not both satisfied for \(21< n\leq 1863\). (The reviewer independently did the latter calculation and found it checked out.) The proof of the new theorem involves very detailed calculations on \(L\) series for a character modulo \(q\) on the field.

MSC:
11R11 Quadratic extensions
11R29 Class numbers, class groups, discriminants
11R42 Zeta functions and \(L\)-functions of number fields
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