is mod statement correct

Question

$2^{2n+1}$%$(2n + 1) = 1$ for all n > 1000.

I am confused to prove if it is wrong or right. As the value of n is very large.

If it is wrong then what would be the correct statement.

Answer 1

No,

Counter example $n=1001$

$$2^{2n+1} \equiv 2\pmod {2n+1}$$