Prove that there exist $135$ consecutive positive integers so that the $n$th least is divisible by a perfect $n$th power greater than $1$

Question

Prove that there exist 135 consecutive positive integers so that the second least is divisible by a perfect square $> 1$, the third least is divisible by a perfect cube $> 1$, the fourth least is divisible by a perfect fourth power $> 1$, and so on.

How should I go about doing this?

I thought perhaps I should use Fermat's little theorem, or its corollary?

Thanks!

Answer 1

Use the Chinese Remainder Theorem. Pick $134$ distinct primes. The perfect square is the square of the first, the cube is the cube of the second, and so on. All your moduli are distinct, so CRT guarantees a solution. If you use the smallest primes in order and $N$ is the least of your $135$ numbers, you have $N+1 \equiv 0 \pmod {2^2}, N+2 \equiv 0 \pmod {3^3}, N+3 \equiv 0 \pmod {5^4}\ldots$

Answer 2

By the Chinese remainder theorem, there is an integer $n$ such that $n\equiv -k\ (\text{mod}\ p_k^{k+1})$ for all $k=1,2,\dots 134$, (where $p_k$ is the $k^\text{th}$ smallest prime). Then $n, n+1, \dots, n+134$ satisfy the required condition.