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Note on Pairs of Consecutive Residues of Polynomials

Published online by Cambridge University Press:  20 November 2018

Kenneth S. Williams
Kenneth S. Williams*
Affiliation:
Carleton University
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Let f(x) be a polynomial of degree d ≥ 3 with integral coefficients, 'say,

In a previous paper [6] I deduced, from a deep result of Lang and Weil [2], that there is a constant k1(d), depending only on d, such that for all primes p ≥ k1(d), p ⫮ ad, f(x) has a pair of consecutive residues (mod p), that i s, there exists an integer r(0 ≤r ≤ p-1) with the property that

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 11 , Issue 1 , April 1968 , pp. 79 - 83
Copyright
Copyright © Canadian Mathematical Society 1968

References

1. Bombieri, E. and Davenport, H., On two problems of Mordell, Amer. J. Math., 88 (1966), 61-70.10.2307/2373047Google Scholar
2. Lang, S. and Weil, A., Number of points of varieties in finite fields, Amer. J. Math., 76(1954), 819-827. 82 10.2307/2372655Google Scholar
3. Mc Cann, K. and Williams, K. S., On the residues of a cubic polynomial (mod p), Canad. Math. Bull., 10(1967), 29-38.10.4153/CMB-1967-004-0Google Scholar
4. McCann, K. and Williams, K. S., The distribution of the residues of a quartic polynomial, Glasgow Math. Journal 8 (1967), 67-88.10.1017/S0017089500000136Google Scholar
5. Tietäväinen, A., On non - residues of a polynomial, Ann. Univ. Turku., Ser. Al, 94 (1966), 3-6.Google Scholar
6. Williams, K.S., Pairs of consecutive residues of polynomials, Can. J. Math., 19(1967), 655-666.10.4153/CJM-1967-060-1Google Scholar