Diophantine Equations and Bernoulli Polynomials

Wiley - 2002
Yu. F. Bilu1, B. Brindza2,3, P. Kirschenhofer4, Á. Pintér3,2, R. F. Tichy4, A. Schinzel5
1A2X, Université Bordeaux 1, 351 cours de la Libération, Talence cedex, France
2University of Debrecen, Hungary
3Department of Mathematics, Debrecen
4Montanuniversität Leoben, Leoben, Austria
5Mathematical Institute PAN, Warszawa, Poland

Tóm tắt

Given m, n ≥ 2, we prove that, for sufficiently large y, the sum 1 n +···+ y n is not a product of m consecutive integers. We also prove that for m ≠ n we have 1 m +···+ x m ≠ 1 n +···+ y n , provided x, y are sufficiently large. Among other auxiliary facts, we show that Bernoulli polynomials of odd index are indecomposable, and those of even index are ‘almost’ indecomposable, a result of independent interest.

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Tài liệu tham khảo

Baker, A.: Bounds for solutions of hyperelliptic equations, Proc. Cambridge Philos. Soc. 65 (1969), 439-444.

Baker, A. and Coates, J.: Integer points on curves of genus 1, Math. Proc. Cambridge Philos. Soc. 67 (1970), 592-602.

Beukers, F., Shorey, T. N. and Tijdeman, R.: Irreducibility of polynomials and arithmetic progressions with equal product of terms, In: K. Győry, H. Iwaniec, J. Urbanowicz (eds), Number Theory in Progress: Proc. Int. Conf. in Number Theory in Honor of A. Schinzel, Zakopane, 1997, W. de Gruyter, 1999, pp. 11-26.

Bilu, Yu. F., Stoll, Th. and Tichy, R. F.: Octahedrons with equally many lattice points, Period. Math. Hungar. 40 (2000), 229-238.

Bilu, Yu. F. and Tichy, R. F.: The Diophantine equation f (x) = g(y), Acta Arith. 95 (2000), 261-288.

Brillhart, J.: On the Euler and Bernoulli polynomials, J. Reine Angew.Math. 234 (1969), 45-64.

Brindza, B.: On some generalizations of the diophantine equation 1k + 2k + ⋯ + xk = yk, Acta Arith. 44 (1984), 99-107.

Brindza, B.: Power values of sums 1k + 2k + ⋯ + xk, Number Theory II (Budapest 1987), Colloq. Math. Soc. János Bolyai 51 (1990), 595-611.

Brindza, B. and Pintér, Á.: On equal values of power sums, Acta Arith. 77 (1996), 303-307.

Brindza, B. and Pintér, Á.: On the irreducibility of some polynomials in two variables, Acta Arith. 82 (1997), 303-307.

Davenport, H., Lewis, D. J. and Schinzel, A.: Equations of the form f (x) = g(y), Quart. J. Math. Oxford 12 (1961), 304-312.

Dujella, A. and Tichy, R. F.: Diophantine equations for second order recursive sequences of polynomials, Quart. J. Math. Oxford 52 (2001), 161-169.

Erdős, P. and Selfridge, J. L.: The product of consecutive integers is never a power, Illinois J. Math. 19 (1975), 292-301.

Fried, M.: On a theorem of Ritt and related Diophantine problems, J. Reine Angew. Math. 264 (1974), 40-55.

Győry, K., Tijdeman, R. and Voorhoeve, M.: On the equation 1k + 2k + ⋯ + xk = yk, Acta Arith. 37 (1980), 234-240.

Hajdu, L. and Pintér, Á.: Combinatorial diophantine equations, Publ. Math. Debrecen 56 (2000), 391-403.

Kano, H.: On the Equation s(1k + 2k + ⋯ + xk) + r = by z, Tokyo J. Math. 13 (1990), 441-448.

Rademacher, H.: Topics in Analytic Number Theory, Springer-Verlag, Berlin, 1973.

Saradha,N., Shorey, T. N. and Tijdeman, R.: On arithmetic progressions of equal length with equal products, Math. Proc. Cambridge Philos. Soc. 117 (1995), 193-201.

Schinzel, A.: Selected Topics on Polynomials, Univ. Michigan Press, Ann Arbor, 1982.

Shorey, T. N. and Tijdeman, R.: Some methods of Erdős applied to cnite arithmetic progressions, The Mathematics of Paul Erdős, Algorithms Combin. 13 (1997), 251-267.

Siegel, C. L.: Über einige Anwendungen Diophantischer Approximationen, Abh. Preuss Akad. Wiss. Phys.-Math. Kl., 1929, Nr. 1; Ges. Abh., Band 1, 209-266.

Voorhoeve, M., Győry, K. and Tijdeman, R.: On the diophantine equation 1k + 2k + ⋯ + xk + R(x) = y z, Acta Math. 143 (1979), 1-8, corrections: Acta Math. 159 (1987), 151.

Yuan, P. Z.: On the special Diophantine equation a x n ) = by y + c, Publ.Math. Debrecen 44 (1994), 137-143.

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