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Abstract Singmaster’s conjecture asserts that every natural number greater than one occurs at most a bounded number of times in Pascal’s triangle; that is, for any natural number $$t \geq 2$$, the number of solutions to the equation $$\binom{n}{m} = t$$ for natural numbers $$1 \leq m \lt n$$ is bounded. In this paper we establish this result in the interior region $$\exp(\log^{2/3+\varepsilon} n) \leq m \leq n - \exp(\log^{2/3+\varepsilon} n)$$ for any fixed ɛ > 0. Indeed, when t is sufficiently large depending on ɛ, we show that there are at most four solutions (or at most two in either half of Pascal’s triangle) in this region. We also establish analogous results for the equation $$(n)_m = t$$, where $$(n)_m := n(n-1) \dots (n-m+1)$$ denotes the falling factorial. 

Abstract We evaluate asymptotically the variance of the number of squarefree integers up to x in short intervals of length $$H < x^{6/11 - \varepsilon }$$ H < x 6 / 11 - ε and the variance of the number of squarefree integers up to x in arithmetic progressions modulo q with $$q > x^{5/11 + \varepsilon }$$ q > x 5 / 11 + ε . On the assumption of respectively the Lindelöf Hypothesis and the Generalized Lindelöf Hypothesis we show that these ranges can be improved to respectively $$H < x^{2/3 - \varepsilon }$$ H < x 2 / 3 - ε and $$q > x^{1/3 + \varepsilon }$$ q > x 1 / 3 + ε . Furthermore we show that obtaining a bound sharp up to factors of $$H^{\varepsilon }$$ H ε in the full range $$H < x^{1 - \varepsilon }$$ H < x 1 - ε is equivalent to the Riemann Hypothesis. These results improve on a result of Hall (Mathematika 29(1):7–17, 1982) for short intervals, and earlier results of Warlimont, Vaughan, Blomer, Nunes and Le Boudec in the case of arithmetic progressions.