My age is a perfect square number of years greater than a perfect square; it is also a perfect square number of years less than a perfect square. Next year my age will be a prime number less than 50. What is my age this year?
For reference, let’s list perfect squares and prime numbers less than 50.
Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81
Primes <50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
If we now look at the ages that are one less than each of the prime numbers, we can quickly check whether that age is a perfect square number of years greater than a perfect square and a perfect square number of years less than a perfect square. We see that 46 does not work and 42 does not work, but 40 does work (4 greater than 36 and 9 less than 49). Are there any other solutions?