Are there infinitely many $m\in\mathbb N$ such that $35\times2^m+1$ is $\textbf{not}$ a prime number?
Thanks in advance.
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Sign up to join this communityAre there infinitely many $m\in\mathbb N$ such that $35\times2^m+1$ is $\textbf{not}$ a prime number?
Thanks in advance.
Yes.
For natural $k$ let $m=4+10k$. Then $35 \times 2^m+1$ is divisible by $11$, since $35 \cdot 2^{4+10k} \equiv -1 \pmod {11}$ and $2^m$ is periodic modulo all primes.
For any positive integer $m$, there are infinitely many $n$ such that $35 \times 2^n+1$ is divisible by $35 \times 2^m+1$. Indeed, it suffices to take $n$ such that $2^{n-m} \equiv 1 \mod 35 \times 2^m+1$.