Zero sums amongst roots and Cilleruelo’s conjecture on the LCM of polynomial sequences Article Swipe
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Marc Technau
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YOU?
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· 2025
· Open Access
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· DOI: https://doi.org/10.1007/s11856-025-2870-7
· OA: W4416971387
YOU?
·
· 2025
· Open Access
·
· DOI: https://doi.org/10.1007/s11856-025-2870-7
· OA: W4416971387
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 Q f ( 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 Q f ( N ).
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