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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

\end{displaymath} (1)

\end{displaymath} (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
$\displaystyle x$ $\textstyle =$ $\displaystyle m^2+n^2$ (3)
$\displaystyle h$ $\textstyle =$ $\displaystyle 4mn(m^2-n^2),$ (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


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.

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© 1996-9 Eric W. Weisstein