Square-free pairs $\mathbf {n^2+n+1}$, $\mathbf {n^2+n+2}$

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2026年3月23日

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S. I. Dimitrov

摘要

In this paper, we prove that there exist infinitely many square-free pairs of the form $n^2+n+1$ $n^{2} + n + 1$ , $n^2+n+2$ $n^{2} + n + 2$ . We also establish an asymptotic formula for the number of such square-free pairs when n does not exceed given sufficiently large positive number. A key point in our proof is the application of bijective correspondence between the number of representations of number by binary quadratic form and the incongruent solutions of quadratic congruence.

Square-free pairs \(\mathbf {n^2+n+1}\), \(\mathbf {n^2+n+2}\)

摘要