This problem was posed by the mathematicians Théodore and Jean de Palerma in a mathematical tournament organized by Frederick II in Pisa in 1225 (Ore 1988, pp. 188-191). A family of solutions is
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, A056203,
A057103, and A057105
in "The On-Line Encyclopedia of Integer Sequences."
such that, when a given positive
integer 
is added or subtracted, new square numbers are obtained
so that
are integers. 
and 
are then given by
, 
, and 
by a positive integer 
, which replaces 
by 
. For example, 
, 
, and 
give 
, 
, 
, and 
. Fibonacci proved that all numbers 
(the congrua) are divisible by
24 (Ore 1988, p. 191). Fermat's
right triangle theorem is equivalent to the result that a congruum
cannot be a square number.
) for small 
and 
(Ore 1988, p. 191). The 
column uses the absolute value
of OEIS A057105.
