## Task 11/78

A mathematician writes to another: "This year I will be 100 years old, next year 200."

Obviously, the age references are not in the decimal system. How old is the mathematician?
Let $x$ be the base of the system for the first age statement, and $y$ the base of the system for the second age statement. To solve the problem, the Diophantine equation $x^{2}+1=2 y^{2}$ or $x^{2}=2 y^{2}-1$ with $x, y \in N, x, y>1$ needs to be solved.

For biological reasons, it is certain that $y<9$ (since otherwise $2 y^{2}=162$, which can be almost certainly ruled out as an age). Therefore, it suffices to examine the numbers $y=2$ to $y=8$ to see if $2 y^{2}-1$ is a perfect square.

It is found that only for $y=5$ a natural number $x$ is a solution to the equation: $x=7$. Thus, the mathematician will be $100_{(7)}=49_{(10)}$ years old this year, and next year $200_{(5)}=50_{(10)}$.