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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
##### Keywords:
class number; quadratic field; Richaud-Degert type
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