Congruum Problem

Find a Square Number $x^2$ such that, when a given number $h$ is added or subtracted, new Square Numbers are obtained so that

$$x^2+h=a^2$$ (1)

and

$$x^2-h=b^2.$$ (2)

This problem was posed by the mathematicians Théodore and Jean de Palerma in a mathematical tournament organized by Frederick II in Pisa in 1225. The solution (Ore 1988, pp. 188-191) is

$$x = m^2+n^2$$ (3)

$$h = 4mn(m^2-n^2),$$ (4)

where $m$ and $n$ are Integers. Fibonacci proved that all numbers $h$ (the Congrua) are divisible by 24. Fermat's Right Triangle Theorem is equivalent to the result that a congruum cannot be a Square Number. A table for small $m$ and $n$ is given in Ore (1988, p. 191), and a larger one (for $h\leq 1000$) by Lagrange (1977).

$m$	$n$	$h$	$x$
2	1	24	5
3	1	96	10
3	2	120	13
4	1	240	17
4	3	336	25

See also Concordant Form, Congruent Numbers, Square Number

References

Alter, R. and Curtz, T. B. ``A Note on Congruent Numbers.'' Math. Comput. 28, 303-305, 1974.

Alter, R.; Curtz, T. B.; and Kubota, K. K. ``Remarks and Results on Congruent Numbers.'' In Proc. Third Southeastern Conference on Combinatorics, Graph Theory, and Computing, 1972, Boca Raton, FL. Boca Raton, FL: Florida Atlantic University, pp. 27-35, 1972.

Bastien, L. ``Nombres congruents.'' Interméd. des Math. 22, 231-232, 1915.

Gérardin, A. ``Nombres congruents.'' Interméd. des Math. 22, 52-53, 1915.

Lagrange, J. ``Construction d'une table de nombres congruents.'' Calculateurs en Math., Bull. Soc. math. France., Mémoire 49-50, 125-130, 1977.

Ore, Ø. Number Theory and Its History. New York: Dover, 1988.