## Another $S$-unit variant of Diophantine tuples

HTML articles powered by AMS MathViewer

- by Clemens Fuchs and Sebastian Heintze PDF
- Proc. Amer. Math. Soc.
**149**(2021), 27-35 Request permission

## Abstract:

We show that there are only finitely many triples of integers $0 < a < b < c$ such that the product of any two of them is the value of a given polynomial with integer coefficients evaluated at an $S$-unit that is also a positive integer. The proof is based on a result of Corvaja and Zannier and thus is ultimately a consequence of the Schmidt subspace theorem.## References

- P. Corvaja and U. Zannier,
*On the greatest prime factor of $(ab+1)(ac+1)$*, Proc. Amer. Math. Soc.**131**(2003), no. 6, 1705–1709. MR**1955256**, DOI 10.1090/S0002-9939-02-06771-0 - Pietro Corvaja and Umberto Zannier,
*A lower bound for the height of a rational function at $S$-unit points*, Monatsh. Math.**144**(2005), no. 3, 203–224. MR**2130274**, DOI 10.1007/s00605-004-0273-0 - A. Dujella, https://web.math.pmf.unizg.hr/~duje/dtuples.html
- Clemens Fuchs, Florian Luca, and Laszlo Szalay,
*Diophantine triples with values in binary recurrences*, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)**7**(2008), no. 4, 579–608. MR**2483637** - Katalin Gyarmati,
*A polynomial extension of a problem of Diophantus*, Publ. Math. Debrecen**66**(2005), no. 3-4, 389–405. MR**2137776** - K. Győry, A. Sárközy, and C. L. Stewart,
*On the number of prime factors of integers of the form $ab+1$*, Acta Arith.**74**(1996), no. 4, 365–385. MR**1378230**, DOI 10.4064/aa-74-4-365-385 - L. Hajdu and A. Sárközy,
*On multiplicative decompositions of polynomial sequences, I*, Acta Arith.**184**(2018), no. 2, 139–150. MR**3841151**, DOI 10.4064/aa170620-13-12 - L. Hajdu and A. Sárközy,
*On multiplicative decompositions of polynomial sequences, II*, Acta Arith.**186**(2018), no. 2, 191–200. MR**3870712**, DOI 10.4064/aa171116-7-3 - W. J. LeVeque,
*On the equation $y^{m}=f(x)$*, Acta Arith.**9**(1964), 209–219. MR**169813**, DOI 10.4064/aa-9-3-209-219 - Florian Luca and Volker Ziegler,
*A note on the number of $S$-Diophantine quadruples*, Commun. Math.**22**(2014), no. 1, 49–55. MR**3233726** - Preda Mihăilescu,
*Primary cyclotomic units and a proof of Catalan’s conjecture*, J. Reine Angew. Math.**572**(2004), 167–195. MR**2076124**, DOI 10.1515/crll.2004.048 - László Szalay and Volker Ziegler,
*On an $S$-unit variant of Diophantine $m$-tuples*, Publ. Math. Debrecen**83**(2013), no. 1-2, 97–121. MR**3081229**, DOI 10.5486/PMD.2013.5521 - László Szalay and Volker Ziegler,
*$S$-Diophantine quadruples with two primes congruent to 3 modulo 4*, Integers**13**(2013), Paper No. A80, 9. MR**3167927** - László Szalay and Volker Ziegler,
*$S$-Diophantine quadruples with $S=\{2,q\}$*, Int. J. Number Theory**11**(2015), no. 3, 849–868. MR**3327847**, DOI 10.1142/S1793042115500475

## Additional Information

**Clemens Fuchs**- Affiliation: Department of Mathematics, University of Salzburg, Hellbrunnerstr. 34, A-5020 Salzburg, Austria
- MR Author ID: 705384
- ORCID: 0000-0002-0304-0775
- Email: clemens.fuchs@sbg.ac.at
**Sebastian Heintze**- Affiliation: Department of Mathematics, University of Salzburg, Hellbrunnerstr. 34, A-5020 Salzburg, Austria
- ORCID: 0000-0002-6356-1986
- Email: sebastian.heintze@sbg.ac.at
- Received by editor(s): October 21, 2019
- Received by editor(s) in revised form: April 20, 2020
- Published electronically: October 16, 2020
- Additional Notes: This research was supported by Austrian Science Fund (FWF): I4406.
- Communicated by: Rachel Pries
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**149**(2021), 27-35 - MSC (2010): Primary 11D61
- DOI: https://doi.org/10.1090/proc/15193
- MathSciNet review: 4172583