Skip to main content Accessibility help

Rational points on Erdős–Selfridge superelliptic curves

  • Michael A. Bennett (a1) and Samir Siksek (a2)


Given $k\geqslant 2$ , we show that there are at most finitely many rational numbers $x$ and $y\neq 0$ and integers $\ell \geqslant 2$ (with $(k,\ell )\neq (2,2)$ ) for which

$$\begin{eqnarray}x(x+1)\cdots (x+k-1)=y^{\ell }.\end{eqnarray}$$
In particular, if we assume that $\ell$ is prime, then all such triples $(x,y,\ell )$ satisfy either $y=0$ or $\ell <\exp (3^{k})$ .



Hide All
[BBGH06] Bennett, M. A., Bruin, N. B., Győry, K. and Hajdu, L., Powers from products of consecutive terms in arithmetic progression , Proc. Lond. Math. Soc. (3) 92 (2006), 273306.
[BD13] Bennett, M. A. and Dahmen, S., Klein forms and the generalized superelliptic equation , Ann. of Math. (2) 177 (2013), 171239.
[BCDT01] Breuil, C., Conrad, B., Diamond, F. and Taylor, R., On the modularity of elliptic curves over ℚ: wild 3-adic exercises , J. Amer. Math. Soc. 14 (2001), 843939.
[DM97] Darmon, H. and Merel, L., Winding quotients and some variants of Fermat’s last theorem , J. Reine Angew. Math. 490 (1997), 81100.
[ES75] Erdős, P. and Selfridge, J. L., The product of consecutive integers is never a power , Illinois J. Math. 19 (1975), 292301.
[Fal83] Faltings, G., Endlichkeitssätze für abelsche Varietäten über Zahlkörpen , Invent. Math. 73 (1983), 349366.
[GHP09] Győry, K., Hajdu, L. and Pintér, Á., Perfect powers from products of consecutive terms in arithmetic progression , Compositio Math. 145 (2009), 845864.
[GHS04] Győry, K., Hajdu, L. and Saradha, N., On the diophantine equation n (n + d)⋯n + (k - 1d) = by l , Canad. Math. Bull. 47 (2004), 373388.
[Kra97] Kraus, A., Majorations effectives pour l’équation de Fermat généralisée , Canad. J. Math. 49 (1997), 11391161.
[LS03] Lakhal, M. and Sander, J. W., Rational points on the superelliptic Erdős–Selfridge curve of fifth degree , Mathematika 50 (2003), 113124.
[Mar05] Martin, G., Dimensions of the spaces of cuspforms and newforms on 𝛤0(N) and 𝛤1(N) , J. Number Theory 112 (2005), 298331.
[Maz78] Mazur, B., Rational isogenies of prime degree , Invent. Math. 44 (1978), 129162.
[Rib90] Ribet, K., On modular representations of Gal(/ℚ) arising from modular forms , Invent. Math. 100 (1990), 431476.
[San99] Sander, J. W., Rational points on a class of superelliptic curves , J. Lond. Math. Soc. (2) 59 (1999), 422434.
[ST76] Schinzel, A. and Tijdeman, R., On the equation y m = P (x) , Acta Arith. XXXI (1976), 199204.
[Sch76] Schoenfeld, L., Sharper bounds for the Chebyshev functions 𝜃(x) and 𝜓(x) II , Math. Comp. 30 (1976), 337360.
[Sik12] Siksek, S., The modular approach to Diophantine equations , in Explicit Methods in Number Theory: Rational Points and Diophantine Equations, Panoramas et Syntheses, vol. 36, eds Belabas, K., Beukers, F., Gaudry, P., McCallum, W., Poonen, B., Siksek, S., Stoll, M. and Watkins, M. (Société Mathématique de France, Paris, 2012), 151179.
MathJax is a JavaScript display engine for mathematics. For more information see


MSC classification

Rational points on Erdős–Selfridge superelliptic curves

  • Michael A. Bennett (a1) and Samir Siksek (a2)


Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed