Question 162007
Ok, the first thing to do is set up your variables.  We will choose 3 consecutive integers using just 1 variable n:

n
n + 1
n + 2

Then, set up the equation:

{{{(n+1)^2+(n+2)^2 = n^2 + 21}}}

And solve:

{{{n^2+2n+1 + n^2+4n+4 = n^2 + 21}}}
{{{2n^2+6n+5 = n^2 + 21}}}
{{{n^2+6n-16 = 0}}}

Then, use the quadratic equation:

{{{(-6 +- sqrt(6^2-4*1*-16))/2}}}
{{{(-6 +- sqrt(100))/2}}}
{{{(-6 +- 10)/2}}}

This gives you an answer of n = 2, -8

So there are 2 solutions:

2, 3, 4

AND

-8, -7, -6