Glasnik Matematicki, Vol. 50, No. 2 (2015), 269-277.

ON TWO OF JOHN LEECH'S UNSOLVED PROBLEMS CONCERNING RATIONAL CUBOIDS

Allan MacLeod

Statistics, O.R. and Mathematics Group, University of the West of Scotland, High St., Paisley,, Scotland. PA1 2BE
e-mail: allan.macleod@uws.ac.uk


Abstract.   Let {X,Y,Z,A,B,C} + be such that X2+Y2=A2, X2+Z2=B2 and Y2+Z2=C2. We consider the problem of finding T + such that either

T2-X2=◻,   T2-Y2=◻,   T2-Z2=◻

or
T2-A2=◻,   T2-B2=◻,   T2-C2=◻.

We show that problem 2 always has a solution and we provide a formula for T. Extensive computation has been unable to find a single solution of problem 1.

2010 Mathematics Subject Classification.   11D09, 11Y50.

Key words and phrases.   Rational cuboid, elliptic curve.


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DOI: 10.3336/gm.50.2.02


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