Congruum Problem
Find a Square Number 
such that, when a given number 
is added or subtracted, new Square Numbers are obtained so that
 | (1) |
 | (2) |
in Pisa in 1225. The solution (Ore 1988, pp. 188-191) is |  |  | (3) |
 | |  | (4) |
where
and 
are Integers. Fibonacci proved that all numbers (the Congrua) are divisible by24. Fermat's Right Triangle Theorem is equivalent to the result that a congruum cannot be a Square Number. Atable for small and is given in Ore (1988, p. 191), and a larger one (for 
) by Lagrange (1977). | | |  |
| 2 | 1 | 24 | 5 |
| 3 | 1 | 96 | 10 |
| 3 | 2 | 120 | 13 |
| 4 | 1 | 240 | 17 |
| 4 | 3 | 336 | 25 |
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.
- Schur's Lemma
- Schur's Problem
- Schur's Representation Lemma
- Schur's Theorem
- Schwarzian Derivative
- Schwarz-Pick Lemma
- Schwarz Reflection Principle
- Schwarz's Inequality
- Schwarz's Triangle Problem
- Schwarz Triangle
- Schwenk's Formula
- Schönflies Symbol
- Schönflies Theorem
- Scientific Notation
- s-Cluster
- Score Sequence
- Screw
- Screw Theorem
- Scruple
- Sea Horse Valley
- Searching
- Search Tree
- Seashell
- Secant
- Secant Line