TOPICS
Search

Congruum Problem

Find a square number x^2 such that, when a given integer 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. a and b are then given by

a=m^2+2mn-n^2
(5)
b=n^2+2mn-m^2.
(6)

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<=1000) by Lagrange (1977). The first

mnhxab
SloaneA057103A055096A057104A057105
2124571
319610142
3212013177
4124017237
4238420284
43336253117

See also

Concordant Form, Congruent Number, Congruum, Square Number

Explore with Wolfram|Alpha

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."

Referenced on Wolfram|Alpha

Congruum Problem

Cite this as:

Weisstein, Eric W. "Congruum Problem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CongruumProblem.html

Subject classifications