COMPLETE SOLUTION OF A PROBLEM OF DIOPHANTUS AND EULER

ANDREJ DUJELLA; CLEMENS FUCHS

doi:10.1112/S002461070400609X

Abstract

It is proved that there does not exist a set of four positive integers with the property that the product of any two of its distinct elements plus their sum is a perfect square. This settles an old problem investigated by Diophantus and Euler.

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Dujella, Andrej Fuchs, Clemens and Walsh, Gary 2006. Diophantine m-tuples for linear polynomials II. Equal degrees. Journal of Number Theory, Vol. 120, Issue. 2, p. 213.

Dujella, Andrej and Franušić, Zrinka 2007. On Differences of Two Squares in Some Quadratic Fields. Rocky Mountain Journal of Mathematics, Vol. 37, Issue. 2,

Filipin, Alan 2009. On the Size of Sets in Which $xy + 4$ is Always a Square. Rocky Mountain Journal of Mathematics, Vol. 39, Issue. 4,

Fujita, Yasutsugu 2009. The Hoggatt-Bergum Conjecture on $D(-1)$-Triples $\{F_{2k+1}, F_{2k+3}, F_{2k+5}\}$ and Integer Points on the Attached Elliptic Curves. Rocky Mountain Journal of Mathematics, Vol. 39, Issue. 6,

Franušić, Zrinka 2009. A diophantine problem in ℤ[1 + \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\sqrt d$ \end{document})/2]. Studia Scientiarum Mathematicarum Hungarica, Vol. 46, Issue. 1, p. 103.

Fujita, Yasutsugu 2009. Extensions of the D(∓k 2)-triples {k 2, k 2 ± 1, 4k 2 ± 1}. Periodica Mathematica Hungarica, Vol. 59, Issue. 1, p. 81.

He, Bo and Togbé, Alain 2009. On the family of diophantine triples {k + 1, 4k, 9k + 3}. Periodica Mathematica Hungarica, Vol. 58, Issue. 1, p. 59.

He, B. and Togbé, A. 2009. On a family of diophantine triples {k, A 2 k + 2A, (A + 1)2 k + 2 (A + 1)} with two parameters. Acta Mathematica Hungarica, Vol. 124, Issue. 1-2, p. 99.

Filipin, Alan and Fujita, Yasutsugu 2011. The D(-k2)-triple {1,k2+1,k2+4} with k prime. Glasnik matematicki, Vol. 46, Issue. 2, p. 311.

He, Bo and Togbé, Alain 2011. On the D(−1)-triple {1,k2+1,k2+2k+2} and its unique D(1)-extension. Journal of Number Theory, Vol. 131, Issue. 1, p. 120.

Narkiewicz, Władysław 2012. Rational Number Theory in the 20th Century. p. 307.

Fujita, Yasutsugu and Togbé, Alain 2012. The extension of the D(−k 2)-pair {k 2,k 2 + 1}. Periodica Mathematica Hungarica, Vol. 65, Issue. 1, p. 75.

Dujella, Andrej and Fuchs, Clemens 2012. On a problem of Diophantus for rationals. Journal of Number Theory, Vol. 132, Issue. 10, p. 2075.

Soldo, Ivan 2013. On the extensibility of D(−1)-triples {1, b, c} in the ring \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\mathbb{Z}\left[ {\sqrt { - t} } \right]$ \end{document}, t > 0. Studia Scientiarum Mathematicarum Hungarica, Vol. 50, Issue. 3, p. 296.

Matthews, Keith R. Robertson, John P. and White, Jim 2013. On a Diophantine equation of Andrej Dujella. Glasnik Matematicki, Vol. 48, Issue. 2, p. 265.

Srinivasan, Anitha 2014. On the prime divisors of elements of a D(-1) quadruple. Glasnik Matematicki, Vol. 49, Issue. 2, p. 275.

Elsholtz, Christian Filipin, Alan and Fujita, Yasutsugu 2014. On Diophantine quintuples and $$D(-1)$$ D ( - 1 ) -quadruples. Monatshefte für Mathematik, Vol. 175, Issue. 2, p. 227.

Zhang, Yifan and Grossman, George 2014. Diophantine Triples and Extendibility of {1, 2, 5} and {1, 5, 10}. The Fibonacci Quarterly, Vol. 52, Issue. 5, p. 212.

Srinivasan, Anitha 2015. D(-1)-quadruples and products of two primes. Glasnik Matematicki, Vol. 50, Issue. 2, p. 261.

Soldo, Ivan 2016. $$D(-1)$$ D ( - 1 ) -triples of the Form $$\{1,b,c\}$$ { 1 , b , c } in the Ring $${\mathbb {Z}}[\sqrt{-t}], t>0$$ Z [ - t ] , t > 0. Bulletin of the Malaysian Mathematical Sciences Society, Vol. 39, Issue. 3, p. 1201.

Download full list

Article contents

COMPLETE SOLUTION OF A PROBLEM OF DIOPHANTUS AND EULER

Abstract

Keywords

Access options

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Article contents

COMPLETE SOLUTION OF A PROBLEM OF DIOPHANTUS AND EULER

Abstract

Keywords

Access options

Save article to Kindle

Save article to Dropbox

Save article to Google Drive

Reply to: Submit a response

Your details

You have entered the maximum number of contributors

Conflicting interests