Glasnik Matematicki, Vol. 47, No. 1 (2012), 31-51.

ON A FAMILY OF TWO-PARAMETRIC D(4)-TRIPLES

Alan Filipin, Bo He and Alain Togbé

Faculty of Civil Engineering, University of Zagreb, Fra Andrije Kačića-Miošića 26, 10000 Zagreb, Croatia
e-mail: filipin@grad.hr

Department of Mathematics, ABa Teacher's College, Wenchuan, Sichuan, 623000, P. R. China
e-mail: bhe@live.cn

Mathematics Department, Purdue University North Central, 1401 S, U.S. 421, Westville IN 46391, USA
e-mail: atogbe@pnc.edu

Abstract.   Let k be a positive integer. In this paper, we study a parametric family of the sets of integers {k,A2k+4A,(A+1)2k+4(A+1),d}. We prove that if d is a positive integer such that the product of any two distinct elements of that set increased by 4 is a perfect square, then

d= (A4 + 2A3 + A2)k3 + (8A3 + 12A2 + 4A)k2 + (20A2 + 20A + 4) k + (16A + 8)
for 1≤ A ≤22 and A ≥ 51767.

2010 Mathematics Subject Classification.   11D09, 11D25, 11J86.

Key words and phrases.   Diophantine m-tuples, Pell equations, Baker's method.

Full text (PDF) (free access)

DOI: 10.3336/gm.47.1.03

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