Prove that there exists infinitely many integers n, such that n,n+1, and n+2 are each the sum of two squares of integers.

Solution:

If we set n=kk+kk, then n+2=(k+1)(k+1)+(k-1)(k-1).

Therefore, it is enough to express n+1=2kk+1 as the sum of two squares for infinitely many k.

We try

2kk+1=(k+a)(k+a)+(k-b)(k-b),

2kk+1=2kk+aa+bb+2k(a-b),

2k(b-a)=bb+aa-1.

Letting b-a=1, we obtain

k=a(a+1).