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<p>We make progress on a conjecture of Cilleruelo on the growth of the least common multiple of consecutive values of an irreducible polynomial <i>f</i> on the additional hypothesis that the polynomial be even. This strengthens earlier work of Rudnick–Maynard and Sah subject to that additional hypothesis when the degree of <i>f</i> exceeds two. The improvement rests upon a different treatment of ‘large’ prime divisors of <i>Q</i><sub><i>f</i></sub>(<i>N</i>) = ∣<i>f</i>(1)⋯<i>f</i>(<i>N</i>)∣ by means of certain zero sums amongst the roots of <i>f</i>. A similar argument was recently used by Baier and Dey with regard to another problem. The same method also allows for further improvements on a related conjecture of Sah on the size of the radical of <i>Q</i><sub><i>f</i></sub>(<i>N</i>).</p>

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Zero sums amongst roots and Cilleruelo’s conjecture on the LCM of polynomial sequences

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  • 发表日期：

    2025年11月30日

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  • Marc Technau

摘要

We make progress on a conjecture of Cilleruelo on the growth of the least common multiple of consecutive values of an irreducible polynomial f on the additional hypothesis that the polynomial be even. This strengthens earlier work of Rudnick–Maynard and Sah subject to that additional hypothesis when the degree of f exceeds two. The improvement rests upon a different treatment of ‘large’ prime divisors of Qf(N) = ∣f(1)⋯f(N)∣ by means of certain zero sums amongst the roots of f. A similar argument was recently used by Baier and Dey with regard to another problem. The same method also allows for further improvements on a related conjecture of Sah on the size of the radical of Qf(N).