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} ℚ+ be such that X2+Y2=A2, X2+Z2=B2 and Y2+Z2=C2. We consider the problem of finding T ℚ+ 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.

Full text (PDF) (free access)

DOI: 10.3336/gm.50.2.02

References:

  1. A. Bremner, The rational cuboid and a quartic surface, Rocky Mountain J. Math. 18 (1988), 105-121.
    MathSciNet     CrossRef

  2. R. K. Guy, Unsolved problems in number theory, 3rd ed., Springer-Verlag, New York, 2004.
    MathSciNet     CrossRef

  3. J. Leech, The rational cuboid revisited, Amer. Math. Monthly 84, (1977), 518-533.
    MathSciNet     CrossRef

  4. R. van Luijk, On perfect cuboids, Undergraduate Thesis, Leiden University, 2000.
  5. L. J. Mordell, Diophantine equations, Academic Press, London, 1969.
    MathSciNet    

  6. J. H. Silverman and J. Tate, Rational points on elliptic curves, Springer-Verlag, New York, 1992.
    MathSciNet     CrossRef

  7. D. Simon, Computing the rank of elliptic curves over number fields, LMS J. Comput. Math. 5 (2002), 7-17.
    MathSciNet     CrossRef
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