โจทย์จาก Elementary Number Theory บางข้อครับ
A1. Show that if $x,y,z$ are positive integers, then $(xy+1)(yz+1)(zx+1)$ is a perfect square if and only if $xy+1,yz+1,zx+1$ are all perfect squares.
A2. Find infinitely many triples $(a,b,c)$ of positive integers such that $a,b,c$ are in arithmetic progression and such that $ab+1,bc+1,$ and $ca+1$ are perfect square.
A4. If $a,b,c$ are positive integers such that $0<a^2+b^2-abc\le c$ show that $a^2+b^2-abc$ is a perfect square.
A10. Let $n$ be a positive integer with $n \ge 3$. Show that
$n^{n^{n^{n}}}-n^{n^{n}}$ is divisible by $1989$