2025 using primes

Puzzle

The number 2025 is not prime. As a matter of fact, it’s a perfect square: $2025 = 452$. You cannot make $2025$ by adding two distinct primes. To do so, you’d have to add an even prime and an odd prime. The only even prime is $2$, but $2025 − 2 = 2023$, which is not prime (it’s equal to $7∙172$). But you can make 2025 by adding three distinct primes. For example, $661 + 673 + 691 = 2025$. You can also make 2025 by adding four distinct primes: $2 + 659 + 673 + 691 = 2025$. What is the greatest number of distinct primes that add up to 2025?

Solution