Generalized Diophantine 𝑚-tuples

Dixit, Anup; Kim, Seoyoung; Murty, M. Ram

doi:10.1090/proc/15776

Generalized Diophantine $m$-tuples
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by Anup B. Dixit, Seoyoung Kim and M. Ram Murty

Proc. Amer. Math. Soc. 150 (2022), 1455-1465

DOI: https://doi.org/10.1090/proc/15776

Published electronically: January 20, 2022

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Abstract:

For non-zero integers $n$ and $k\geq 2$, a generalized Diophantine $m$-tuple with property $D_k(n)$ is a set of $m$ positive integers $\{a_1,a_2,\ldots , a_m\}$ such that $a_ia_j + n$ is a $k$-th power for $1\leq i< j\leq m$. Define $M_k(n)≔\sup \{|S| : S$ has property $D_k(n)\}$. In this paper, we study upper bounds on $M_k(n)$, as we vary $n$ over positive integers. In particular, we show that for $k\geq 3$, $M_k(n)$ is $O(\log n)$ and further assuming the Paley graph conjecture, $M_k(n)$ is $O((\log n)^{\epsilon })$. The problem for $k=2$ was studied by a long list of authors that goes back to Diophantus.

References

Joseph Arkin, V. E. Hoggatt Jr., and E. G. Straus, On Euler’s solution of a problem of Diophantus, Fibonacci Quart. 17 (1979), no. 4, 333–339. MR 550175
A. Baker and H. Davenport, The equations $3x^{2}-2=y^{2}$ and $8x^{2}-7=z^{2}$, Quart. J. Math. Oxford Ser. (2) 20 (1969), 129–137. MR 248079, DOI 10.1093/qmath/20.1.129
Riley Becker and M. Ram Murty, Diophantine $m$-tuples with the property $D(n)$, Glas. Mat. Ser. III 54(74) (2019), no. 1, 65–75. MR 3964398, DOI 10.3336/gm.54.1.05
Attila Bérczes, Andrej Dujella, Lajos Hajdu, and Florian Luca, On the size of sets whose elements have perfect power $n$-shifted products, Publ. Math. Debrecen 79 (2011), no. 3-4, 325–339. MR 2907969, DOI 10.5486/PMD.2011.5155
Yann Bugeaud and Andrej Dujella, On a problem of Diophantus for higher powers, Math. Proc. Cambridge Philos. Soc. 135 (2003), no. 1, 1–10. MR 1990827, DOI 10.1017/S0305004102006588
Lucia Caporaso, Joe Harris, and Barry Mazur, Uniformity of rational points, J. Amer. Math. Soc. 10 (1997), no. 1, 1–35. MR 1325796, DOI 10.1090/S0894-0347-97-00195-1
Alina Carmen Cojocaru and M. Ram Murty, An introduction to sieve methods and their applications, London Mathematical Society Student Texts, vol. 66, Cambridge University Press, Cambridge, 2006. MR 2200366
Andrej Dujella, On the size of Diophantine $m$-tuples, Math. Proc. Cambridge Philos. Soc. 132 (2002), no. 1, 23–33. MR 1866322, DOI 10.1017/S0305004101005515
Andrej Dujella, There are only finitely many Diophantine quintuples, J. Reine Angew. Math. 566 (2004), 183–214. MR 2039327, DOI 10.1515/crll.2004.003
J.-H. Evertse, On the quantitative subspace theorem, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 377 (2010), no. Issledovaniya po Teorii Chisel. 10, 217–240, 245; English transl., J. Math. Sci. (N.Y.) 171 (2010), no. 6, 824–837. MR 2753657, DOI 10.1007/s10958-010-0185-6
J.-H. Evertse and J. H. Silverman, Uniform bounds for the number of solutions to $Y^n=f(X)$, Math. Proc. Cambridge Philos. Soc. 100 (1986), no. 2, 237–248. MR 848850, DOI 10.1017/S0305004100066068
G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), no. 3, 349–366 (German). MR 718935, DOI 10.1007/BF01388432
P. X. Gallagher, A larger sieve, Acta Arith. 18 (1971), 77–81. MR 291120, DOI 10.4064/aa-18-1-77-81
Katalin Gyarmati, On a problem of Diophantus, Acta Arith. 97 (2001), no. 1, 53–65. MR 1819622, DOI 10.4064/aa97-1-3
Ahmet M. Güloğlu and M. Ram Murty, The Paley graph conjecture and Diophantine $m$-tuples, J. Combin. Theory Ser. A 170 (2020), 105155, 9. MR 4016095, DOI 10.1016/j.jcta.2019.105155
Bo He, Alain Togbé, and Volker Ziegler, There is no Diophantine quintuple, Trans. Amer. Math. Soc. 371 (2019), no. 9, 6665–6709. MR 3937341, DOI 10.1090/tran/7573
Wolfgang M. Schmidt, Integer points on curves of genus $1$, Compositio Math. 81 (1992), no. 1, 33–59. MR 1145607
Igor E. Shparlinski, Finite fields: theory and computation, Mathematics and its Applications, vol. 477, Kluwer Academic Publishers, Dordrecht, 1999. The meeting point of number theory, computer science, coding theory and cryptography. MR 1745660, DOI 10.1007/978-94-015-9239-0
Joseph H. Silverman and John Tate, Rational points on elliptic curves, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992. MR 1171452, DOI 10.1007/978-1-4757-4252-7
I. M. Vinogradov, Elements of number theory, Dover Publications, Inc., New York, 1954. Translated by S. Kravetz. MR 0062138