×

On a Diophantine equation involving quadratic characters. (English) Zbl 0584.10008

The equation of the title is (*) \(\sum^{xf}_{a=1}\chi (a)a^ n=by^ z\), where \(n\) and \(b\) are integers and \(\chi\) is a primitive quadratic residue class character with conductor \(f\). The author’s Theorem 1: If \(b\neq 0\) and \(n\) is sufficiently large, then (*) has only finitely many solutions in integers \(x,y\geq 1\) and \(z\geq 2\), with effective upper bounds for \(x,y,z\). In Theorems 2 and 3 the author makes the assumption about \(n\) more explicit, providing that \(\chi (-1)=-1\) and \(f\) is of a certain type, e.g., \(f\) is a prime \(\equiv 3\pmod 8\) or, simply, \(f=4\). In the former case it is enough to require that \(n\geq 3\) and \(4\nmid n\), in the latter case that either \(n=3\) or \(n\geq 6\).
The proofs follow ideas developed by K. Györy, R. Tijdeman and M. Voorhoeve in case of the principal character \(\chi =1\) [Acta Math. 143, 1–8 (1979; Zbl 0426.10019) and Acta Arith. 37, 233–240 (1980; Zbl 0439.10010)]. The author’s main contribution is the study of analytic properties and zeros of generalized Bernoulli polynomials or, rather, of certain related polynomials. In fact, a known identity relates the left hand side of (*) to such a polynomial.
The paper is concluded with a generalization of Theorem 1. This is analogous to a result in the first paper cited above; it concerns the equation obtained from (*) by adding a polynomial \(R_ n(x)\) with integer coefficients.

MSC:

11D61 Exponential Diophantine equations
11B68 Bernoulli and Euler numbers and polynomials
PDFBibTeX XMLCite
Full Text: Numdam EuDML

References:

[1] B.C. Berndt : Character analogues of the Poisson and Euler-MacLaurin summation formulas with applications , J. Number Theory 7 (1975) 413-445. · Zbl 0316.10023 · doi:10.1016/0022-314X(75)90045-1
[2] J. Brillhart : On the Euler and Bernoulli polynomials , J. Reine Angew. Math. 234 (1969) 45-64. · Zbl 0167.35401 · doi:10.1515/crll.1969.234.45
[3] B. Brindza : On S-integral solutions of the equation ym = f(x) , Acta Math. Hung. 44 (1984) 133-139. · Zbl 0552.10009 · doi:10.1007/BF01974110
[4] L. Carlitz : A conjecture concerning the Euler numbers , Amer. Math. Monthly 69 (1962) 538-540. · Zbl 0105.26403 · doi:10.2307/2311198
[5] K. Dilcher : Irreducibilty of generalized Bernoulli polynomials , preprint. · Zbl 0558.10012
[6] K. Dilcher : Asymptotic behaviour of Bernoulli, Euler, and generalzed Bernoulli polynomials , to appear in J. Approx. Theory. · Zbl 0609.10008 · doi:10.1016/0021-9045(87)90071-2
[7] R.J. Duffin : Algorithms for localizing roots of a polynomial and the Pisot Vijayaraghavan numbers , Pacific J. Math. 74 (1978) 47-56. · Zbl 0381.12001 · doi:10.2140/pjm.1978.74.47
[8] R. Ernvall : Generalized Bernoulli numbers, generalized irregular primes, and class numbers , Ann. Univ. Turku, Ser. AI, 178 (1979) 72 p. · Zbl 0403.12010
[9] K. Györy , R. Tijdeman and M. Voorhoeve : On the equation 1k + 2k + \cdot \cdot \cdot + xk = yz , Acta Arith. 37 (1980), 233-240. · Zbl 0365.10014
[10] K. Iwasawa : Lectures on p-acid L-functions , Princeton University Press (1972). · Zbl 0236.12001 · doi:10.1515/9781400881703
[11] D.E. Knuth And T.J. Buckholtz : Computation of tangent. Euler, and Bernoulli numbers . Math. Comp. 21 (1967) 663-688. · Zbl 0178.04401 · doi:10.2307/2005010
[12] L.J. Mordell : Diophantine equations , Academic Press, London (1969). · Zbl 0188.34503
[13] N. Nörlund : Vorlesungen über Differenzenrechung , Springer Verlag, Berlin (1924). · JFM 50.0315.02
[14] G. Polya and G. Szegö : Aufgaben und Lehrsätze aus der Analysis , Springer Verlag, Berlin (1925). · JFM 51.0173.01
[15] J.J. Schäffer : The equation 1p + 2p + 3p + \cdot \cdot \cdot + np = mq , Acta Math. 95 (1956) 155-189. · Zbl 0071.03702 · doi:10.1007/BF02401100
[16] M. Voorhoeve , K. Györy and R. Tijdeman : On the diophantine equation 1k + 2k + \cdot \cdot \cdot + xk + R(x) = yz , Acta Math. 143 (1979) 1-8. · Zbl 0426.10019 · doi:10.1007/BF02392086
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.