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A note on the Hermite–Rankin constant

Kazuomi Sawatani, Takao Watanabe, Kenji Okuda (2010)

Journal de Théorie des Nombres de Bordeaux

We generalize Poor and Yuen’s inequality to the Hermite–Rankin constant γ n , k and the Bergé–Martinet constant γ n , k . Moreover, we determine explicit values of some low- dimensional Hermite–Rankin and Bergé–Martinet constants by applying Rankin’s inequality and some inequalities proven by Bergé and Martinet to explicit values of γ 5 , γ 7 , γ 4 , 2 and γ n ( n 8 ).

A note on the number of S -Diophantine quadruples

Florian Luca, Volker Ziegler (2014)

Communications in Mathematics

Let ( a 1 , , a m ) be an m -tuple of positive, pairwise distinct integers. If for all 1 i < j m the prime divisors of a i a j + 1 come from the same fixed set S , then we call the m -tuple S -Diophantine. In this note we estimate the number of S -Diophantine quadruples in terms of | S | = r .

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