Zero sums amongst roots and Cilleruelo's conjecture on the LCM of polynomial sequences
Abstract: 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 $Q_f(N) = f(1)\cdots f(N)$ 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 $Q_f(N)$.
- S. Baier and P. K. Dey. Prime powers dividing products of consecutive integer values of x2n+1superscript𝑥superscript2𝑛1x^{2^{n}}+1italic_x start_POSTSUPERSCRIPT 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT + 1. Res. Number Theory, 6(7):1–12, 2020. doi: 10.1007/s40993-019-0182-x.
- J. Cilleruelo. The least common multiple of a quadratic sequence. Compos. Math., 147(4):1129–1150, 2011. doi: 10.1112/S0010437X10005191.
- Equidistribution of roots of a quadratic congruence to prime moduli. Ann. Math. (2), 141(2):423–441, 1995. doi: 10.2307/2118527.
- A. Entin and S. Landsberg. The least common multiple of polynomial values over function fields, 2023. Preprint: https://arxiv.org/abs/2310.04164.
- E. Leumi. The LCM problem for function fields. Master’s thesis, Tel Aviv University, Tel Aviv, Israel, April 2021.
- J. Maynard and Z. Rudnick. A lower bound on the least common multiple of polynomial sequences. Riv. Mat. Univ. Parma (N.S.), 12(1):143–150, 2021.
- J. Merikoski. On the largest prime factor of n2+1superscript𝑛21n^{2}+1italic_n start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 1. J. Eur. Math. Soc. (JEMS), 25(4):1253–1284, 2023. doi: 10.4171/JEMS/1216.
- A. Sah. An improved bound on the least common multiple of polynomial sequences. J. Théor. Nombres Bordx., 32(3):891–899, 2020. doi: 10.5802/jtnb.1146.
- Á. Tóth. Roots of quadratic congruences. Int. Math. Res. Not., 2000(14):719–739, 2000. doi: 10.1155/S1073792800000404.
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