Primes can be represented as a^2-b^2=(a+b)(a-b) if we choose a-b=1 and a+b=p so
a=(p+1)/2 and b=(p-1)/2 so a and b are consecutive integers one odd and another
even and so all odd numbers (and thus primes too) can be represented as
difference of consecutive squares:

1^2-0^2=1
2^2-1^2=3
3^2-2^2=5
4^2-3^2=7

If we sum them we get:
1+3=2^2
1+3+5=3^2
1+3+5+7=4^2
The sum of the first n odd integers (beginning with 1) is n^2 (1)

If 1+....+2n-1=n^2 then 1+....+2n-1+2n+1=n^2+2n+1=(n+1)^2
(1) is proved by induction

And any prime can be represented as a difference of squares
((p+1)/2)^2-((p-1)/2)^2 while only 1 mod 4 primes can be represented as a sum of
squares (all in N)