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SMOOTH VALUES OF POLYNOMIALS

  • J. W. BOBER (a1) (a2), D. FRETWELL (a1) (a3), G. MARTIN (a4) and T. D. WOOLEY (a5)
Abstract

Given $f\in \mathbb{Z}[t]$ of positive degree, we investigate the existence of auxiliary polynomials $g\in \mathbb{Z}[t]$ for which $f(g(t))$ factors as a product of polynomials of small relative degree. One consequence of this work shows that for any quadratic polynomial $f\in \mathbb{Z}[t]$ and any $\unicode[STIX]{x1D700}>0$ , there are infinitely many $n\in \mathbb{N}$ for which the largest prime factor of $f(n)$ is no larger than $n^{\unicode[STIX]{x1D700}}$ .

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The third author’s work is partially supported by a National Sciences and Engineering Research Council of Canada Discovery Grant. The fourth author’s work is supported by a European Research Council Advanced Grant under the European Union’s Horizon 2020 research and innovation programme via grant agreement no. 695223.

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[1] Balog, A. and Wooley, T. D., ‘On strings of consecutive integers with no large prime factors’, J. Aust. Math. Soc. Ser. A 64(2) (1998), 266276.
[2] Bhargava, M., ‘Most hyperelliptic curves over have no rational points’, Preprint, 2013, available at arXiv:1308:0395.
[3] Bhargava, M., Gross, B. H. and Wang, X., ‘A positive proportion of locally soluble hyperelliptic curves over ℚ have no point over any odd degree extension’ (with an appendix by T. Dokchitser and V. Dokchitser), J. Amer. Math. Soc. 30(2) (2017), 451493.
[4] Dartyge, C., Martin, G. and Tenenbaum, G., ‘Polynomial values free of large prime factors’, Period. Math. Hungar. 43(1–2) (2001), 111119.
[5] Granville, A. and Pleasants, P., ‘Aurifeuillian factorization’, Math. Comp. 75(253) (2006), 497508.
[6] Harrington, J., ‘On the factorization of the trinomials x n + cx n-1 + d ’, Int. J. Number Theory 8(6) (2012), 15131518.
[7] Ljunggren, W., ‘On the irreducibility of certain trinomials and quadrinomials’, Math. Scand. 8 (1960), 6570.
[8] Martin, G., ‘An asymptotic formula for the number of smooth values of a polynomial’, J. Number Theory 93(2) (2002), 108182.
[9] Schinzel, A., ‘On two theorems of Gelfond and some of their applications’, Acta Arith. 13 (1967), 177236.
[10] Schinzel, A., Polynomials with Special Regard to Reducibility, Encyclopedia of Mathematics and its Applications, 77 (Cambridge University Press, Cambridge, 2000).
[11] Selmer, E. S., ‘On the irreducibility of certain trinomials’, Math. Scand. 4 (1956), 287302.
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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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