Congruum Problem

Find a square number  such that, when a given integer  is added or subtracted, new square numbers are obtained so that

(1)

and

(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

			(3)
			(4)

where  and  are integers.  and  are then given by

			(5)
			(6)

Fibonacci proved that all numbers  (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  and  is given in Ore (1988, p. 191), and a larger one (for ) by Lagrange (1977). The first

Sloane		A057103	A055096	A057104	A057105
2	1	24	5	7	1
3	1	96	10	14	2
3	2	120	13	17	7
4	1	240	17	23	7
4	2	384	20	28	4
4	3	336	25	31	17

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.

Sloane, N. J. A. Sequences A055096, A057103, A057104, and A057105 in "The On-Line Encyclopedia of Integer Sequences."