466. What are two square numbers whose difference is 133?

## Montferrier's Problems.
466. Henri Monde, a peasant from a village near the city of Tours. In 1840, he was presented to the president of the Paris Academy of Sciences, the famous French mathematician Poncelet, by the owner of a boarding school in this city. With Poncelet's permission, the assembled members of the Academy asked the boy two questions: what is the square of 756 and how many minutes are in 52 years. He immediately gave the correct answers. After this, a commission was organized, which, as stated, included the most prominent mathematicians. A report on the examination was prepared for the meeting on December 14, 1840, by Augustin Cauchy. When Monde was given this problem, he immediately provided one solution: 66 and 67. Then he was told that there was a solution in simpler numbers. After a moment of thought, he said that the second solution was: 6 and 13. Monde's method is based on the application of the identity:

$$
x^{2}-y^{2}=(x-y)^{2}-2 y(y-x)=(x-y)^{2}+2 y(x-y)
$$

i.e., the question reduces to subtracting such a square from the number 133 so that the remainder is divisible by twice the square root. Testing the squares $1,4,9,16,25,36,49, \ldots$, we see that only 1 and 49 satisfy this condition:

$$
\frac{133-1}{2}=66 ; \frac{133-49}{14}=6
$$

The usual method is as follows:

$$
(x+y)(x-y)=1 \cdot 133=19 \cdot 7
$$

Monferrand, A. S., a French mathematician, author of several mathematical textbooks, who published in three volumes the "Dictionnaire des sciences mathématiques pures et appliquées" (the 2nd edition was published in 1838).