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The least common multiple and lattice points on hyperbolas. (English) Zbl 0983.11058

Let \(a_1a_2,\dots, a_k\) be \(k\) given integers with \(X\leq a_1< a_2<\dots< a_k\leq X+L\). The main result of the paper is an estimation for the lower bound of the least common multiple \(\text{LCM} [a_1,a_2,\dots, a_k]\). From this, one can derive an estimation for the number of lattice points on a short arc of a hyperbola. Assume that \(k\) distinct lattice points \((a_i,b_i)\) are lying on the hyperbola \(ab=N\) with \(a_1< a_2<\dots< a_k\). Then a lower bound for \(a_k-a_1\) is given.

MSC:

11P21 Lattice points in specified regions
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