Abstract
It is known that under the Dirichlet product, the set of arithmetic functions in several variables becomes a unique factorization domain. A. Zaharescu and M. Zaki proved an analog of the ABC conjecture in this ring and showed that there exists a non-trivial solution to the Fermat equation \(z^n=x^n+y^n\) (\(n\ge 3\)). It is also known that under the Cauchy product, the set of arithmetic functions becomes a unique factorization domain. In this paper, we consider the ring of arithmetic functions in several variables under the Cauchy product and prove an analog of the ABC conjecture in this ring to show that there exists a non-trivial solution to the Fermat equation \(z^n=x^n+y^n\) (\(n\ge 3\)).
Similar content being viewed by others
References
Alkan, E., Zaharescu, A., Zaki, M.: Arithmetical functions in several variables. Int. J. Number Theory 1, 383–399 (2005)
Alkan, E., Zaharescu, A., Zaki, M.: Unitary convolution for arithmetical functions in several variables. Hiroshima Math. J. 36, 113–124 (2006)
Cashwell, E.D., Everett, C.J.: The ring of number-theoretic functions. Pac. J. Math. 9, 975–985 (1959)
Lang, S.: Old and new conjectured diophantine inequalities. Bull. Am. Math. Soc. 23, 37–75 (1990)
Lang, S.: Algebra, revised 3 edn. Springer, Berlin (2002)
Mason, R.C.: Diophantine Equations over Function Fields, London Mathematical Society Lecture Note Series, vol. 96. Cambridge University Press, Cambridge (1984)
Masser, D.: Open problems. In: Chen, W.W.L. (ed.) Proceedings Symposium on Analytic Number Theory. Imperial College, London (1985)
Mihăilescu, P.: Primary cyclotomic units and a proof of Catalan’s conjecture. J. Reine Angew. Math. 572, 167–195 (2004)
Oesterlé, J.: Nouvelles approches du “théorème” de Fermat. In: Séminaire Bourbaki 1987/88, Astèrisque, vol. 161–162, pp. 165–186 (1989)
Sivaramakrishnan, R.: Classical Theory of Arithmetic Functions. Marcel Dekker, New York (1989)
Stothers, W.W.: Polynomial identities and Hauptmoduln. Q. J. Math. 32, 349–370 (1981)
Wiles, A.: Modular elliptic curves and Fermat’s last theorem. Ann. Math. 141, 443–551 (1995)
Zaharescu, A., Zaki, M.: Factorization in certain rings of arithmetical functions. Kumamoto J. Math. 21, 29–39 (2008)
Zaharescu, A., Zaki, M.: An ABC analog for arithmetical functions. J. Ramanujan Math. Soc. 25, 345–354 (2010)
Acknowledgements
The author would like to thank the referee for the helpful comments that improved this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Hamahata, Y. Arithmetic functions and the Cauchy product. Arch. Math. 114, 41–50 (2020). https://doi.org/10.1007/s00013-019-01384-9
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-019-01384-9