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- W1982210080 abstract "We first prove that if $a$ has a prime factor not dividing $b$ then there are infinitely many positive integers $n$ such that $binom {an+bn} {an}$ is not divisible by $bn+1$. This confirms a recent conjecture of Z.-W. Sun. Moreover, we provide some new divisibility properties of binomial coefficients: for example, we prove that $binom {12n} {3n}$ and $binom {12n} {4n}$ are divisible by $6n-1$, and that $binom {330n} {88n}$ is divisible by $66n-1$, for all positive integers $n$. As we show, the latter results are in fact consequences of divisibility and positivity results for quotients of $q$-binomial coefficients by $q$-integers, generalizing the positivity of $q$-Catalan numbers. We also put forward several related conjectures." @default.
- W1982210080 created "2016-06-24" @default.
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- W1982210080 date "2014-02-01" @default.
- W1982210080 modified "2023-10-16" @default.
- W1982210080 title "Some divisibility properties of binomial and q -binomial coefficients" @default.
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- W1982210080 doi "https://doi.org/10.1016/j.jnt.2013.08.012" @default.
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