Abstract
If $k$ is a sufficiently large positive integer, we show that the Diophantine equation $$ n ( n+d) \cdots (n+ (k-1)d ) = y^{\ell } $$ has at most finitely many solutions in positive integers $n, d, y$ and $\ell $, with gcd$(n,d)=1$ and $\ell \geq 2$. Our proof relies upon Frey-Hellegouarch curves and results on supersingular primes for elliptic curves without complex multiplication, derived from upper bounds for short character sums and sieves, analytic and combinatorial.
-
[BBGH]
M. A. Bennett, N. Bruin, K. GyHory, and L. Hajdu, "Powers from products of consecutive terms in arithmetic progression," Proc. London Math. Soc. (3), vol. 92, iss. 2, pp. 273-306, 2006.@ARTICLE{BBGH,
author = {Bennett, M. A. and Bruin, N. and Gy{ő}ry, K. and Hajdu, L.},
title = {Powers from products of consecutive terms in arithmetic progression},
journal = {Proc. London Math. Soc. (3)},
fjournal = {Proceedings of the London Mathematical Society. Third Series},
volume = {92},
year = {2006},
number = {2},
pages = {273--306},
issn = {0024-6115},
mrclass = {11D61 (11B25)},
mrnumber = {2205718},
mrreviewer = {Natarajan Saradha},
doi = {10.1112/S0024611505015625},
url = {https://doi.org/10.1112/S0024611505015625},
zblnumber = {1178.11033},
} -
[EBPAP] M. A. Bennett, G. Martin, K. O’Bryant, and A. Rechnitzer, "Explicit bounds for primes in arithmetic progressions," Illinois J. Math., vol. 62, iss. 1-4, pp. 427-532, 2018.
@ARTICLE{EBPAP,
author = {Bennett, Michael A. and Martin, Greg and O'Bryant, Kevin and Rechnitzer, Andrew},
title = {Explicit bounds for primes in arithmetic progressions},
journal = {Illinois J. Math.},
fjournal = {Illinois Journal of Mathematics},
volume = {62},
year = {2018},
number = {1-4},
pages = {427--532},
issn = {0019-2082},
mrclass = {11N13 (11M20 11M26 11N37 11Y35 11Y40)},
mrnumber = {3922423},
mrreviewer = {Timothy S. Trudgian},
doi = {10.1215/ijm/1552442669},
url = {https://doi.org/10.1215/ijm/1552442669},
zblnumber = {07036793},
} -
[BeSi] M. A. Bennett and S. Siksek, "Rational points on Erdős-Selfridge superelliptic curves," Compos. Math., vol. 152, iss. 11, pp. 2249-2254, 2016.
@ARTICLE{BeSi,
author = {Bennett, Michael A. and Siksek, Samir},
title = {Rational points on {E}rd{ő}s-{S}elfridge superelliptic curves},
journal = {Compos. Math.},
fjournal = {Compositio Mathematica},
volume = {152},
year = {2016},
number = {11},
pages = {2249--2254},
issn = {0010-437X},
mrclass = {11D61 (11D41 11F41 11F80)},
mrnumber = {3577894},
mrreviewer = {B. Sury},
doi = {10.1112/S0010437X16007569},
url = {https://doi.org/10.1112/S0010437X16007569},
zblnumber = {1407.11081},
} -
[BennettSkinner] M. A. Bennett and C. M. Skinner, "Ternary Diophantine equations via Galois representations and modular forms," Canad. J. Math., vol. 56, iss. 1, pp. 23-54, 2004.
@ARTICLE{BennettSkinner,
author = {Bennett, Michael A. and Skinner, Chris M.},
title = {Ternary {D}iophantine equations via {G}alois representations and modular forms},
journal = {Canad. J. Math.},
fjournal = {Canadian Journal of Mathematics. Journal Canadien de Math{é}matiques},
volume = {56},
year = {2004},
number = {1},
pages = {23--54},
issn = {0008-414X},
mrclass = {11D41 (11F11 11F80)},
mrnumber = {2031121},
mrreviewer = {Henri Darmon},
doi = {10.4153/CJM-2004-002-2},
url = {https://doi.org/10.4153/CJM-2004-002-2},
zblnumber = {1053.11025},
} -
[BPR] Y. Bilu, P. Parent, and M. Rebolledo, "Rational points on $X^+_0(p^r)$," Ann. Inst. Fourier (Grenoble), vol. 63, iss. 3, pp. 957-984, 2013.
@ARTICLE{BPR,
author = {Bilu, Yuri and Parent, Pierre and Rebolledo, Marusia},
title = {Rational points on {$X^+_0(p^r)$}},
journal = {Ann. Inst. Fourier (Grenoble)},
fjournal = {Universit{é} de Grenoble. Annales de l'Institut Fourier},
volume = {63},
year = {2013},
number = {3},
pages = {957--984},
issn = {0373-0956},
mrclass = {11G18 (11G05 11G16)},
mrnumber = {3137477},
doi = {10.5802/aif.2781},
url = {https://doi.org/10.5802/aif.2781},
zblnumber = {1307.11075},
} -
[Bombieri] E. Bombieri, "A note on the large sieve," Acta Arith., vol. 18, pp. 401-404, 1971.
@ARTICLE{Bombieri,
author = {Bombieri, E.},
title = {A note on the large sieve},
journal = {Acta Arith.},
fjournal = {Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica},
volume = {18},
year = {1971},
pages = {401--404},
issn = {0065-1036},
mrclass = {10.64},
mrnumber = {0286773},
mrreviewer = {W. G. H. Schaal},
doi = {10.4064/aa-18-1-401-404},
url = {https://doi.org/10.4064/aa-18-1-401-404},
zblnumber = {0219.10055},
} -
[Bo] E. Bombieri, "Le grand crible dans la théorie analytique des nombres," Astérisque, iss. 18, p. 103, 1987.
@ARTICLE{Bo,
author = {Bombieri, Enrico},
title = {Le grand crible dans la th{é}orie analytique des nombres},
edition = {Second},
journal = {Ast{é}risque},
fjournal = {Ast{é}risque},
publisher = {Soc. Math. France, Paris},
number = {18},
year = {1987},
pages = {103 pp.},
issn = {0303-1179},
mrclass = {11N35},
mrnumber = {0891718},
mrreviewer = {H.-E. Richert},
zblnumber = {0618.10042},
url = {http://www.numdam.org/item/AST_1987__18__1_0/},
} -
[BreuilConradDiamondTaylor01] C. Breuil, B. Conrad, F. Diamond, and R. Taylor, "On the modularity of elliptic curves over $\Bbb Q$: wild 3-adic exercises," J. Amer. Math. Soc., vol. 14, iss. 4, pp. 843-939, 2001.
@ARTICLE{BreuilConradDiamondTaylor01,
author = {Breuil, Christophe and Conrad, Brian and Diamond, Fred and Taylor, Richard},
title = {On the modularity of elliptic curves over {$\bold Q$}: wild 3-adic exercises},
journal = {J. Amer. Math. Soc.},
fjournal = {Journal of the Amer. Math. Soc.},
volume = {14},
year = {2001},
number = {4},
pages = {843--939},
issn = {0894-0347},
mrclass = {11G05 (11F80 11G07 14G35)},
mrnumber = {1839918},
mrreviewer = {Karl Rubin},
doi = {10.1090/S0894-0347-01-00370-8},
url = {https://doi.org/10.1090/S0894-0347-01-00370-8},
zblnumber = {0982.11033},
} -
[DaGr] H. Darmon and A. Granville, "On the equations $z^m=F(x,y)$ and $Ax^p+By^q=Cz^r$," Bull. London Math. Soc., vol. 27, iss. 6, pp. 513-543, 1995.
@ARTICLE{DaGr,
author = {Darmon, Henri and Granville, Andrew},
title = {On the equations {$z^m=F(x,y)$} and {$Ax^p+By^q=Cz^r$}},
journal = {Bull. London Math. Soc.},
fjournal = {The Bulletin of the London Mathematical Society},
volume = {27},
year = {1995},
number = {6},
pages = {513--543},
issn = {0024-6093},
mrclass = {11D41},
mrnumber = {1348707},
mrreviewer = {Nigel Boston},
doi = {10.1112/blms/27.6.513},
url = {https://doi.org/10.1112/blms/27.6.513},
zblnumber = {0838.11023},
} -
[DM] H. Darmon and L. Merel, "Winding quotients and some variants of Fermat’s last theorem," J. Reine Angew. Math., vol. 490, pp. 81-100, 1997.
@ARTICLE{DM,
author = {Darmon, Henri and Merel, Loïc},
title = {Winding quotients and some variants of {F}ermat's last theorem},
journal = {J. Reine Angew. Math.},
fjournal = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
volume = {490},
year = {1997},
pages = {81--100},
issn = {0075-4102},
mrclass = {11G18 (11D41 11F80 11G05)},
mrnumber = {1468926},
mrreviewer = {Kenneth Kramer},
zblnumber = {0976.11017},
doi = {10.1515/crll.1997.490.81},
url = {https://doi.org/10.1515/crll.1997.490.81},
} -
[Elk] N. D. Elkies, "Distribution of supersingular primes," in Journées Arithmétiques, 1989, Soc. Math. France, Paris, 1991, pp. 127-132 (1992).
@INCOLLECTION{Elk,
author = {Elkies, Noam D.},
title = {Distribution of supersingular primes},
booktitle = {Journ{é}es Arithm{é}tiques, 1989},
venue = {Luminy, 1989},
series = {Ast{é}risque},
fjournal = {Ast{é}risque},
publisher = {Soc. Math. France, Paris},
number = {198-200},
year = {1991},
pages = {127--132 (1992)},
issn = {0303-1179},
mrclass = {11G05},
mrnumber = {1144318},
mrreviewer = {Joseph H. Silverman},
zblnumber = {0754.14019},
url = {http://www.numdam.org/item/AST_1991__198-199-200__127_0/},
} -
[Er55] P. ErdHos, "On the product of consecutive integers. III," Nederl. Akad. Wet., Proc. Ser. A., vol. 58, pp. 85-90, 1955.
@ARTICLE{Er55,
author = {Erd{{ő}}s, P.},
title = {On the product of consecutive integers. {III}},
journal = {Nederl. Akad. Wet., Proc. Ser. A.},
volume = {58},
year = {1955},
pages = {85--90},
mrclass = {10.0X},
mrnumber = {0067915},
mrreviewer = {H. W. Brinkmann},
zblnumber = {0068.03704},
} -
[ErSe] P. ErdHos and J. L. Selfridge, "The product of consecutive integers is never a power," Illinois J. Math., vol. 19, pp. 292-301, 1975.
@ARTICLE{ErSe,
author = {Erd{{ő}}s, P. and Selfridge, J. L.},
title = {The product of consecutive integers is never a power},
journal = {Illinois J. Math.},
fjournal = {Illinois Journal of Mathematics},
volume = {19},
year = {1975},
pages = {292--301},
issn = {0019-2082},
mrclass = {10B05 (10A30)},
mrnumber = {0376517},
mrreviewer = {T. M. Apostol},
zblnumber = {0295.10017},
doi = {10.1215/ijm/1256050816},
url = {https://doi.org/10.1215/ijm/1256050816},
} -
[EST] P. Erdös, C. L. Stewart, and R. Tijdeman, "Some diophantine equations with many solutions," Compositio Math., vol. 66, iss. 1, pp. 37-56, 1988.
@ARTICLE{EST,
author = {Erd{ö}s, P. and Stewart, C. L. and Tijdeman, R.},
title = {Some diophantine equations with many solutions},
journal = {Compositio Math.},
fjournal = {Compositio Mathematica},
volume = {66},
year = {1988},
number = {1},
pages = {37--56},
issn = {0010-437X},
mrclass = {11D41},
mrnumber = {0937987},
mrreviewer = {Takashi Agoh},
url = {http://www.numdam.org/item?id=CM_1988__66_1_37_0},
zblnumber = {0639.10014},
} -
[GrRi] S. W. Graham and C. J. Ringrose, "Lower bounds for least quadratic nonresidues," in Analytic Number Theory (Allerton Park, IL, 1989), Birkhäuser Boston, Boston, MA, 1990, vol. 85, pp. 269-309.
@INCOLLECTION{GrRi,
author = {Graham, S. W. and Ringrose, C. J.},
title = {Lower bounds for least quadratic nonresidues},
booktitle = {Analytic {N}umber {T}heory ({A}llerton {P}ark, {IL},
1989)},
series = {Progr. Math.},
volume = {85},
pages = {269--309},
publisher = {Birkhäuser Boston, Boston, MA},
year = {1990},
mrclass = {11N69 (11L26 11M26)},
mrnumber = {1084186},
mrreviewer = {D. R. Heath-Brown},
zblnumber = {0719.11006},
doi = {10.1007/978-1-4612-3464-7_18},
url = {https://doi.org/10.1007/978-1-4612-3464-7_18},
} -
[Gr] A. Granville, Personal communication.
@MISC{Gr,
author = {Granville, A.},
title = {personal communication},
zblnumber = {},
} -
[Gy] K. Györy, "Power values of products of consecutive integers and binomial coefficients," in Number Theory and its Applications, Kluwer Acad. Publ., Dordrecht, 1999, vol. 2, pp. 145-156.
@INCOLLECTION{Gy,
author = {Györy, K.},
title = {Power values of products of consecutive integers and binomial coefficients},
booktitle = {Number Theory and its Applications},
venue = {{K}yoto, 1997},
series = {Dev. Math.},
volume = {2},
pages = {145--156},
publisher = {Kluwer Acad. Publ., Dordrecht},
year = {1999},
mrclass = {11D61 (11B65)},
mrnumber = {1738813},
mrreviewer = {Natarajan Saradha},
zblnumber = {1074.11502},
} -
[GHP] K. GyHory, L. Hajdu, and Á. Pintér, "Perfect powers from products of consecutive terms in arithmetic progression," Compos. Math., vol. 145, iss. 4, pp. 845-864, 2009.
@ARTICLE{GHP,
author = {Gy{ő}ry, K. and Hajdu, L. and Pint{é}r, \'{A}.},
title = {Perfect powers from products of consecutive terms in arithmetic progression},
journal = {Compos. Math.},
fjournal = {Compositio Mathematica},
volume = {145},
year = {2009},
number = {4},
pages = {845--864},
issn = {0010-437X},
mrclass = {11D61 (11B25)},
mrnumber = {2521247},
mrreviewer = {Michael A. Bennett},
doi = {10.1112/S0010437X09004114},
url = {https://doi.org/10.1112/S0010437X09004114},
zblnumber = {1194.11043},
} -
[IK] H. Iwaniec and E. Kowalski, Analytic Number Theory, Amer. Math. Soc., Providence, RI, 2004, vol. 53.
@BOOK{IK,
author = {Iwaniec, Henryk and Kowalski, Emmanuel},
title = {Analytic Number Theory},
series = {Amer. Math. Soc. Colloq. Publ.},
volume = {53},
publisher = {Amer. Math. Soc., Providence, RI},
year = {2004},
pages = {xii+615},
isbn = {0-8218-3633-1},
mrclass = {11-02 (11Fxx 11Lxx 11Mxx 11Nxx)},
mrnumber = {2061214},
mrreviewer = {K. Soundararajan},
doi = {10.1090/coll/053},
url = {https://doi.org/10.1090/coll/053},
zblnumber = {1059.11001},
} -
[KW1] C. Khare and J. Wintenberger, "Serre’s modularity conjecture. I," Invent. Math., vol. 178, iss. 3, pp. 485-504, 2009.
@ARTICLE{KW1,
author = {Khare, Chandrashekhar and Wintenberger, Jean-Pierre},
title = {Serre's modularity conjecture. {I}},
journal = {Invent. Math.},
fjournal = {Inventiones Mathematicae},
volume = {178},
year = {2009},
number = {3},
pages = {485--504},
issn = {0020-9910},
mrclass = {11F80 (11F11 11F33 11R39)},
mrnumber = {2551763},
mrreviewer = {Gabor Wiese},
doi = {10.1007/s00222-009-0205-7},
url = {https://doi.org/10.1007/s00222-009-0205-7},
zblnumber = {1304.11041},
} -
[KW2] C. Khare and J. Wintenberger, "Serre’s modularity conjecture. II," Invent. Math., vol. 178, iss. 3, pp. 505-586, 2009.
@ARTICLE{KW2,
author = {Khare, Chandrashekhar and Wintenberger, Jean-Pierre},
title = {Serre's modularity conjecture. {II}},
journal = {Invent. Math.},
fjournal = {Inventiones Mathematicae},
volume = {178},
year = {2009},
number = {3},
pages = {505--586},
issn = {0020-9910},
mrclass = {11F80 (11F11 11F33 11R39)},
mrnumber = {2551764},
mrreviewer = {Gabor Wiese},
doi = {10.1007/s00222-009-0206-6},
url = {https://doi.org/10.1007/s00222-009-0206-6},
zblnumber = {1304.11042},
} -
[Koblitz] N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, New York, 1984, vol. 97.
@BOOK{Koblitz,
author = {Koblitz, Neal},
title = {Introduction to Elliptic Curves and Modular Forms},
series = {Grad. Texts in Math.},
volume = {97},
publisher = {Springer-Verlag, New York},
year = {1984},
pages = {viii+248},
isbn = {0-387-96029-5},
mrclass = {11G05 (11G40 14G25 14K07)},
mrnumber = {0766911},
mrreviewer = {Leslie Jane Federer},
doi = {10.1007/978-1-4684-0255-1},
url = {https://doi.org/10.1007/978-1-4684-0255-1},
zblnumber = {0804.11039},
} -
[Kraus] A. Kraus, "Majorations effectives pour l’équation de Fermat généralisée," Canad. J. Math., vol. 49, iss. 6, pp. 1139-1161, 1997.
@ARTICLE{Kraus,
author = {Kraus, Alain},
title = {Majorations effectives pour l'{é}quation de {F}ermat g{é}n{é}ralis{é}e},
journal = {Canad. J. Math.},
fjournal = {Canadian Journal of Mathematics. Journal Canadien de Math{é}matiques},
volume = {49},
year = {1997},
number = {6},
pages = {1139--1161},
issn = {0008-414X},
mrclass = {11D41 (11F33 11F80 11G05)},
mrnumber = {1611640},
mrreviewer = {Kenneth Kramer},
doi = {10.4153/CJM-1997-056-2},
url = {https://doi.org/10.4153/CJM-1997-056-2},
zblnumber = {0908.11017},
} -
[LaSh2] S. Laishram and T. N. Shorey, "Perfect powers in arithmetic progressions," J. Comb. Number Theory, vol. 7, iss. 2, pp. 95-110, 2015.
@ARTICLE{LaSh2,
author = {Laishram, Shanta and Shorey, Tarlok N.},
title = {Perfect powers in arithmetic progressions},
journal = {J. Comb. Number Theory},
fjournal = {Journal of Combinatorics and Number Theory},
volume = {7},
year = {2015},
number = {2},
pages = {95--110},
issn = {1942-5600},
mrclass = {11D61 (11D41)},
mrnumber = {3537553},
mrreviewer = {Szabolcs Tengely},
zblnumber = {1386.11062},
} -
[Lemos] P. Lemos, "Serre’s uniformity conjecture for elliptic curves with rational cyclic isogenies," Trans. Amer. Math. Soc., vol. 371, iss. 1, pp. 137-146, 2019.
@ARTICLE{Lemos,
author = {Lemos, Pedro},
title = {Serre's uniformity conjecture for elliptic curves with rational cyclic isogenies},
journal = {Trans. Amer. Math. Soc.},
fjournal = {Transactions of the Amer. Math. Soc.},
volume = {371},
year = {2019},
number = {1},
pages = {137--146},
issn = {0002-9947},
mrclass = {11G05 (14H52)},
mrnumber = {3885140},
mrreviewer = {\'{A}lvaro Lozano-Robledo},
doi = {10.1090/tran/7198},
url = {https://doi.org/10.1090/tran/7198},
zblnumber = {06993229},
} -
[Lio] J. Liouville, "Sur le produit $m(m+1)(m+2)…(m+n-1)$," J. Math. Pures Appl., vol. 2, pp. 277-278, 1857.
@ARTICLE{Lio,
author={Liouville, J.},
title={Sur le produit $m(m+1)(m+2)…(m+n-1)$},
journal={J. Math. Pures Appl.},
volume={2},
year={1857},
pages={277--278},
url={https://gallica.bnf.fr/ark:/12148/bpt6k164012/f285n2.capture},
} -
[Mars] R. Marszałek, "On the product of consecutive elements of an arithmetic progression," Monatsh. Math., vol. 100, iss. 3, pp. 215-222, 1985.
@ARTICLE{Mars,
author = {Marsza{\l}ek, R.},
title = {On the product of consecutive elements of an arithmetic progression},
journal = {Monatsh. Math.},
fjournal = {Monatshefte für Mathematik},
volume = {100},
year = {1985},
number = {3},
pages = {215--222},
issn = {0026-9255},
mrclass = {11B25},
mrnumber = {0812613},
mrreviewer = {D. A. Klarner},
doi = {10.1007/BF01299269},
url = {https://doi.org/10.1007/BF01299269},
zblnumber = {0582.10011},
} -
[Ma] G. Martin, "Dimensions of the spaces of cusp forms and newforms on $\Gamma_0(N)$ and $\Gamma_1(N)$," J. Number Theory, vol. 112, iss. 2, pp. 298-331, 2005.
@ARTICLE{Ma,
author = {Martin, Greg},
title = {Dimensions of the spaces of cusp forms and newforms on {$\Gamma_0(N)$} and {$\Gamma_1(N)$}},
journal = {J. Number Theory},
fjournal = {Journal of Number Theory},
volume = {112},
year = {2005},
number = {2},
pages = {298--331},
issn = {0022-314X},
mrclass = {11F11 (11F25)},
mrnumber = {2141534},
mrreviewer = {Amir Akbary},
doi = {10.1016/j.jnt.2004.10.009},
url = {https://doi.org/10.1016/j.jnt.2004.10.009},
zblnumber = {1095.11026},
} -
[Mazur] B. Mazur, "Rational isogenies of prime degree (with an appendix by D. Goldfeld)," Invent. Math., vol. 44, iss. 2, pp. 129-162, 1978.
@ARTICLE{Mazur,
author = {Mazur, B.},
title = {Rational isogenies of prime degree (with an appendix by {D}. {G}oldfeld)},
journal = {Invent. Math.},
fjournal = {Inventiones Mathematicae},
volume = {44},
year = {1978},
number = {2},
pages = {129--162},
issn = {0020-9910},
mrclass = {14K07 (10D35 14G25)},
mrnumber = {0482230},
mrreviewer = {V. V. Shokurov},
doi = {10.1007/BF01390348},
url = {https://doi.org/10.1007/BF01390348},
zblnumber = {0386.14009},
} -
[Pl] D. J. Platt, "Numerical computations concerning the GRH," Math. Comp., vol. 85, iss. 302, pp. 3009-3027, 2016.
@ARTICLE{Pl,
author = {Platt, David J.},
title = {Numerical computations concerning the {GRH}},
journal = {Math. Comp.},
fjournal = {Mathematics of Computation},
volume = {85},
year = {2016},
number = {302},
pages = {3009--3027},
issn = {0025-5718},
mrclass = {11M26 (11M06 11P32)},
mrnumber = {3522979},
mrreviewer = {Temenoujka P. Peneva},
doi = {10.1090/mcom/3077},
url = {https://doi.org/10.1090/mcom/3077},
zblnumber = {1345.11064},
} -
[Rah] M. Rahman, Roth’s theorem on $3$-term arithmetic progressions.
@MISC{Rah,
author = {Rahman, M.},
title = {Roth's theorem on $3$-term arithmetic progressions},
url = {https://pdfs.semanticscholar.org/34d5/e5d802d1107b68f1aa76dff994e8b23341c2.pdf},
zblnumber = {},
} -
[RamareRumely] O. Ramaré and R. Rumely, "Primes in arithmetic progressions," Math. Comp., vol. 65, iss. 213, pp. 397-425, 1996.
@ARTICLE{RamareRumely,
author = {Ramar{é},
Olivier and Rumely, Robert},
title = {Primes in arithmetic progressions},
journal = {Math. Comp.},
fjournal = {Mathematics of Computation},
volume = {65},
year = {1996},
number = {213},
pages = {397--425},
issn = {0025-5718},
mrclass = {11N13 (11Y35)},
mrnumber = {1320898},
mrreviewer = {Claudia A. Spiro-Silverman},
doi = {10.1090/S0025-5718-96-00669-2},
url = {https://doi.org/10.1090/S0025-5718-96-00669-2},
zblnumber = {0856.11042},
} -
[Ribet-1990] K. A. Ribet, "On modular representations of ${ Gal}(\overline{\bf Q}/{\bf Q})$ arising from modular forms," Invent. Math., vol. 100, iss. 2, pp. 431-476, 1990.
@ARTICLE{Ribet-1990,
author = {Ribet, K. A.},
title = {On modular representations of {${\rm Gal}(\overline{\bf Q}/{\bf Q})$} arising from modular forms},
journal = {Invent. Math.},
fjournal = {Inventiones Mathematicae},
volume = {100},
year = {1990},
number = {2},
pages = {431--476},
issn = {0020-9910},
mrclass = {11G18 (11F32 11F80 11S37)},
mrnumber = {1047143},
mrreviewer = {Glenn Stevens},
doi = {10.1007/BF01231195},
url = {https://doi.org/10.1007/BF01231195},
zblnumber = {0773.11039},
} -
[RoSc] B. J. Rosser and L. Schoenfeld, "Approximate formulas for some functions of prime numbers," Illinois J. Math., vol. 6, pp. 64-94, 1962.
@ARTICLE{RoSc,
author = {Rosser, J. Barkley and Schoenfeld, Lowell},
title = {Approximate formulas for some functions of prime numbers},
journal = {Illinois J. Math.},
fjournal = {Illinois Journal of Mathematics},
volume = {6},
year = {1962},
pages = {64--94},
issn = {0019-2082},
mrclass = {10.42},
mrnumber = {0137689},
mrreviewer = {B. K. Ghosh},
zblnumber = {0122.05001},
doi = {10.1215/ijm/1255631807},
url = {https://doi.org/10.1215/ijm/1255631807},
} -
[SerreIHES] . J-P. Serre, "Quelques applications du théorème de densité de Chebotarev," Inst. Hautes Études Sci. Publ. Math., vol. 54, pp. 323-401, 1981.
@ARTICLE{SerreIHES,
author = {Serre, {\relax J-P}},
title = {Quelques applications du th{é}orème de densit{é} de {C}hebotarev},
journal = {Inst. Hautes \'{E}tudes Sci. Publ. Math.},
fjournal = {Institut des Hautes \'{E}tudes Scientifiques. Publications Math{é}matiques},
volume = {54},
year = {1981},
pages = {323--401},
issn = {0073-8301},
mrclass = {12A75 (10D99 10H25 14G25)},
mrnumber = {0644559},
mrreviewer = {J. Tunnell},
url = {http://archive.numdam.org/article/PMIHES_1981__54__123_0.pdf},
zblnumber = {0496.12011},
} -
[SerreDuke] . J-P. Serre, "Sur les représentations modulaires de degré $2$ de ${ Gal}(\overline{\bf Q}/{\bf Q})$," Duke Math. J., vol. 54, iss. 1, pp. 179-230, 1987.
@ARTICLE{SerreDuke,
author = {Serre, {\relax J-P}},
title = {Sur les repr{é}sentations modulaires de degr{é} {$2$} de {${\rm Gal}(\overline{\bf Q}/{\bf Q})$}},
journal = {Duke Math. J.},
fjournal = {Duke Mathematical Journal},
volume = {54},
year = {1987},
number = {1},
pages = {179--230},
issn = {0012-7094},
mrclass = {11F11 (11G05 14G15 14G25 14K15)},
mrnumber = {0885783},
mrreviewer = {M. A. Kenku},
doi = {10.1215/S0012-7094-87-05413-5},
url = {https://doi.org/10.1215/S0012-7094-87-05413-5},
zblnumber = {0641.10026},
} -
[Sc] L. Schoenfeld, "Sharper bounds for the Chebyshev functions $\theta (x)$ and $\psi (x)$. II," Math. Comp., vol. 30, iss. 134, pp. 337-360, 1976.
@ARTICLE{Sc,
author = {Schoenfeld, Lowell},
title = {Sharper bounds for the {C}hebyshev functions {$\theta (x)$} and {$\psi (x)$}. {II}},
journal = {Math. Comp.},
fjournal = {Mathematics of Computation},
volume = {30},
year = {1976},
number = {134},
pages = {337--360},
issn = {0025-5718},
mrclass = {10H05},
mrnumber = {0457374},
mrreviewer = {R. P. Brent},
doi = {10.2307/2005976},
url = {https://doi.org/10.2307/2005976},
zblnumber = {0326.10037},
} -
[Sho0] T. N. Shorey, "Some exponential Diophantine equations," in New Advances in Transcendence Theory, Cambridge Univ. Press, Cambridge, 1988, pp. 352-365.
@INCOLLECTION{Sho0,
author = {Shorey, T. N.},
title = {Some exponential {D}iophantine equations},
booktitle = {New Advances in Transcendence Theory},
venue = {{D}urham, 1986},
pages = {352--365},
publisher = {Cambridge Univ. Press, Cambridge},
year = {1988},
mrclass = {11D61 (11J87)},
mrnumber = {0972011},
mrreviewer = {Daniel Bertrand},
zblnumber = {},
} -
[Sho00] T. N. Shorey, "Perfect powers in products of arithmetical progressions with fixed initial term," Indag. Math. (N.S.), vol. 7, iss. 4, pp. 521-525, 1996.
@ARTICLE{Sho00,
author = {Shorey, T. N.},
title = {Perfect powers in products of arithmetical progressions with fixed initial term},
journal = {Indag. Math. (N.S.)},
fjournal = {Koninklijke Nederlandse Akademie van Wetenschappen. Indagationes Mathematicae. New Series},
volume = {7},
year = {1996},
number = {4},
pages = {521--525},
issn = {0019-3577},
mrclass = {11D41 (11B25)},
mrnumber = {1620124},
mrreviewer = {A. Peth{ő}},
doi = {10.1016/S0019-3577(97)89137-9},
url = {https://doi.org/10.1016/S0019-3577(97)89137-9},
zblnumber = {0874.11034},
} -
[Sho1] T. N. Shorey, "Diophantine approximations, Diophantine equations, transcendence and applications," Indian J. Pure Appl. Math., vol. 37, iss. 1, pp. 9-39, 2006.
@ARTICLE{Sho1,
author = {Shorey, T. N.},
title = {Diophantine approximations, {D}iophantine equations, transcendence and applications},
journal = {Indian J. Pure Appl. Math.},
fjournal = {Indian Journal of Pure and Applied Mathematics},
volume = {37},
year = {2006},
number = {1},
pages = {9--39},
issn = {0019-5588},
mrclass = {11J04 (11D45 11D61 11J86)},
mrnumber = {2254063},
mrreviewer = {Roland Qu\^{e}me},
zblnumber = {1207.11074},
} -
[ShTi] T. N. Shorey and R. Tijdeman, "Perfect powers in products of terms in an arithmetical progression," Compositio Math., vol. 75, iss. 3, pp. 307-344, 1990.
@ARTICLE{ShTi,
author = {Shorey, T. N. and Tijdeman, R.},
title = {Perfect powers in products of terms in an arithmetical progression},
journal = {Compositio Math.},
fjournal = {Compositio Mathematica},
volume = {75},
year = {1990},
number = {3},
pages = {307--344},
issn = {0010-437X},
mrclass = {11D61 (11B25)},
mrnumber = {1070417},
mrreviewer = {A. Peth{ő}},
url = {http://www.numdam.org/item?id=CM_1990__75_3_307_0},
zblnumber = {0708.11021},
} -
[Siksek] S. Siksek, "The modular approach to Diophantine equations," in Explicit Methods in Number Theory, Soc. Math. France, Paris, 2012, vol. 36, pp. 151-179.
@INCOLLECTION{Siksek,
author = {Siksek, Samir},
title = {The modular approach to {D}iophantine equations},
booktitle = {Explicit Methods in Number Theory},
series = {Panor. Synthèses},
volume = {36},
pages = {151--179},
publisher = {Soc. Math. France, Paris},
year = {2012},
mrclass = {11D41 (11D61 11G05 11G07)},
mrnumber = {3098134},
mrreviewer = {Nikos Tzanakis},
zblnumber = {1343.11042},
doi = {10.1007/978-0-387-49894-2_7},
url = {https://doi.org/10.1007/978-0-387-49894-2_7},
} -
[Strom] K. R. Stromberg, Introduction to Classical Real Analysis, Wadsworth International, Belmont, Calif., 1981.
@BOOK{Strom,
author = {Stromberg, Karl R.},
title = {Introduction to Classical Real Analysis},
series = {Wadsworth Internat. Math. Ser.},
publisher = {Wadsworth International, Belmont, Calif.},
year = {1981},
pages = {ix+575},
isbn = {0-534-98012-0},
mrclass = {26-01 (26A42 42-01)},
mrnumber = {0604364},
mrreviewer = {R. P. Boas, Jr.},
zblnumber = {0454.26001},
} -
[Tenenbaum] G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Cambridge Univ. Press, Cambridge, 1995, vol. 46.
@BOOK{Tenenbaum,
author = {Tenenbaum, G{é}rald},
title = {Introduction to Analytic and Probabilistic Number Theory},
series = {Cambridge Stud. Adv. Math.},
volume = {46},
note = {translated from the second French edition (1995) by C. B. Thomas},
publisher = {Cambridge Univ. Press, Cambridge},
year = {1995},
pages = {xvi+448},
isbn = {0-521-41261-7},
mrclass = {11-02 (11Kxx 11Mxx 11Nxx)},
mrnumber = {1342300},
mrreviewer = {H. G. Diamond},
zblnumber = {0831.11001},
} -
[Varnavides] P. Varnavides, "On certain sets of positive density," J. London Math. Soc., vol. 34, pp. 358-360, 1959.
-
[Wi] A. Wiles, "Modular elliptic curves and Fermat’s Last Theorem," Ann. of Math. (2), vol. 141, iss. 3, pp. 443-551, 1995.
@ARTICLE{Wi,
author = {Wiles, Andrew},
title = {Modular elliptic curves and {F}ermat's {L}ast {T}heorem},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {141},
year = {1995},
number = {3},
pages = {443--551},
issn = {0003-486X},
mrclass = {11G05 (11D41 11F11 11F80 11G18)},
mrnumber = {1333035},
mrreviewer = {Karl Rubin},
doi = {10.2307/2118559},
url = {https://doi.org/10.2307/2118559},
zblnumber = {0823.11029},
}
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