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This section of Resonance presents thought-provoking questions, and discusses answers a few months later. Readers are invited to send new questions, solutions to old ones and comments, to ‘Think It Over’, Resonance, Indian Academy of Sciences, Bangalore 560 080. Items illustrating ideas and concepts will generally be chosen.

A Problem in Number Theory

M N Deshpande

Find a way of generating infinitely many triples (a,b,c) of positive integers,
with the condition that
ab+1, bc+1 and ca+1 are perfect squares,
and (a+b)/c = 2/3.

Solution:

First we show how to generate infinitely many triples (a,b,c) of positive integers such that
ab+1 , bc+1 and ca+1 are all squares.

Let a,b,c be chosen so that ab+1 is a square, say x^2 ; let c = a+b+2x . ...

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Address for Correspondence
M N Deshpande
p, 22 Panchdeep Nagar
Wardha Road, Nagpur 440 025
Email: dpratap@nagpur.dot.net.in

Indian Academy of Sciences
C.V.Raman Avenue, Post Box No. 8005,
Sadashivanagar Post, Bangalore 560 080