n-tuples of positive integers with the same sum and the same product

Authors:
Yong Zhang and Tianxin Cai

Journal:
Math. Comp. **82** (2013), 617-623

MSC (2010):
Primary 11D25; Secondary 11D72, 11G05

Published electronically:
May 8, 2012

MathSciNet review:
2983039

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, by using the theory of elliptic curves, we prove that for every , there exists infinitely many primitive sets of -tuples of positive integers with the same sum and the same product.

**1.**J. G. Mauldon, Lorraine L. Foster, and Gabriel Robins,*Problems and Solutions: Solutions of Elementary Problems: E2872*, Amer. Math. Monthly**89**(1982), no. 7, 499–500. MR**1539997****2.**Richard K. Guy,*Unsolved problems in number theory*, 3rd ed., Problem Books in Mathematics, Springer-Verlag, New York, 2004. MR**2076335****3.**Desmond MacHale, Gideon Schwarz, W. G. Leavitt, and J. G. Mauldon,*Problems and Solutions: Elementary Problems: E2869-E2872*, Amer. Math. Monthly**88**(1981), no. 2, 147–148. MR**1539609**, 10.2307/2321140**4.**A. Schinzel,*Triples of positive integers with the same sum and the same product*, Serdica Math. J.**22**(1996), no. 4, 587–588. MR**1483607****5.**Joseph H. Silverman and John Tate,*Rational points on elliptic curves*, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992. MR**1171452****6.**T. Skolem, Diophantische Gleichungen, Chelsea, 1950.

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Additional Information

**Yong Zhang**

Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou 310027, People’s Republic of China

Email:
zhangyongzju@163.com

**Tianxin Cai**

Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou 310027, People’s Republic of China

Email:
txcai@zju.edu.cn

DOI:
https://doi.org/10.1090/S0025-5718-2012-02609-3

Keywords:
$n$-tuples,
the system of equations,
elliptic curves

Received by editor(s):
April 20, 2011

Received by editor(s) in revised form:
August 19, 2011

Published electronically:
May 8, 2012

Additional Notes:
This research was supported by China National Science Foundation Grant No.10871169.

Article copyright:
© Copyright 2012
American Mathematical Society