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
Every congruum is obtained from this family by scaling , , 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.
The table gives unscaled solutions () for small and (Ore 1988, p. 191). The column uses the absolute value
of OEIS A057105.
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.
