Congruum Problem
المؤلف:
Alter, R. and Curtz, T. B.
المصدر:
"A Note on Congruent Numbers." Math. Comput. 28
الجزء والصفحة:
...
18-5-2020
1869
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
where 
and 
are integers. 
and 
are then given by
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."
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