# Is 2501 prime?

While walking the dog I encountered the license plate “A 2501”, and because I’m a weirdo my immediate question was to wonder if 2501 is prime.

It’s not a multiple of 3… or 7… or 13.

So, let’s give you some spoiler space, to try the problem without any techmology. Is 2501 prime, or not?

This spoiler space is sponsored by the Illustrative Mathematics Grades 6-8 curriculum, coming very soon! It will be CC-BY, which means you can download and use it free for any purpose, forever.

Enough space yet?

So, having failed to find a simple multiple that divides 2501, I noticed it was one more than a multiple of 4, and a sum of squares:

$2501 = 50^2 + 1^2$

This is a sign that it might be prime, because all primes that are one more than a multiple of 4 can be written as the sum of two squares (try 17 … 21 … 29). But it’s also a sign that it might not be prime, because all primes that are one more than a multiple of 4 can be written as the sum of two squares in exactly one way (try 25 … 65).

So the search changed from finding a factor to finding another sum of squares, and the search ended quickly after remembering 2401 was a perfect square:

$2501 = 49^2 + 10^2$

What’s interesting here is we know from this that 2501 is not prime, but we don’t know what its factors are! Having tested and not finding any prime factors less than 17, this is enough to know that 2501 is the product of exactly two prime factors.

But which ones? To do this, I tried to find a way to write 2501 as a difference of squares. And again, the search ended much more quickly than I expected:

$2501 = 51^2 - 10^2$

$51^2 - 10^2 = (51 + 10)(51 - 10)$

It made me wonder if there are other numbers with the same property, where the number is 1 more than a perfect square, and also has an equal gap to the next square up and next square down… guess what.

26 between 16 and 36

37 between 25 and 49

$n^2 + 1$ between $(n-1)^2$ and $(n+1)^2$

Let me know if you find anything interesting here, or if you have any favorite numerical calculations like these.