Question 79
An integer $n$ is called friendly if there are positive integers $a,b,c$ with $a\lt b\lt c$ and $n=a^2+b^2-c^2$. For example, 1 and 2 are friendly because $1=4^2+7^2-8^2$ and $2=5^2+11^2-12^2$.
a) Verify that $(3x+1)^2+(4x+2)^2-(5x+2)^2=2x+1$ for every $x$.
b) Find integers $m,n$ such that $(3x-m)^2+(4x-n)^2-(5x-5)^2=2x$ for every $x$.
c) Show that 4 is friendly.
d) Show that every positive integer is friendly.
a) Verify that $(3x+1)^2+(4x+2)^2-(5x+2)^2=2x+1$ for every $x$.
b) Find integers $m,n$ such that $(3x-m)^2+(4x-n)^2-(5x-5)^2=2x$ for every $x$.
c) Show that 4 is friendly.
d) Show that every positive integer is friendly.