I'm trying to prove that there exists any m > 0 consecutive non-prime-powers. Where a prime power is $2^2$ or $5^3$. One set of consecutive nonprime powers is 33,34,35,36
I know that the def of CRT requires that the moduli be coprime. I also know that if a number is not a prime power, its divisible by at least 2 distinct primes. So based on that I set up (all ps are distinct)
$$n+0≡0(mod (p_0p_1))$$ $$n+1≡0(mod(p_2p_3))$$ $$n+2≡0(mod(p_4p_5))$$ $$...n+(m-1)≡0(mod(p_{2m-2}p_{2m-1}))$$ $$...n≡-m-1(mod(p_{2m-2}p_{2m-1}))$$
Is it enough to just say that CRT guarantees a solution at this point?
That feels too simple
Also, for guidance, I saw a similar question here
But was a little worried because mine looks a bit different.