#### 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 *X*^{2}+Y^{2}=A^{2}, *X*^{2}+Z^{2}=B^{2} and *Y*^{2}+Z^{2}=C^{2}.
We consider the problem of finding *T ℚ*^{+} such that either

*T*^{2}-X^{2}=◻, T^{2}-Y^{2}=◻, T^{2}-Z^{2}=◻

or

*T*^{2}-A^{2}=◻, T^{2}-B^{2}=◻, T^{2}-C^{2}=◻.

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)** (access from subscribing institutions only)
DOI: 10.3336/gm.50.2.02

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