math190 quiz

quiz 已過，是時候收拾心情準備其他科。
原諒我記憶力有限，以下題目大體題意和原問題相同。

Problem 1. Show that there exist 2010 consecutive positive integers which are divisible by a sq number (not necessaily the same), where sq number is of the form $m^2$, with $m$ being an integer.

Problem 2. Let $x,y\in\mathbb{N}$, find all the ordered pair(s) $(x,y)$ satisfying
$y^2-(x+1)2^x=1$

with proof. (Hint given: There is only one solution)