Question 55
For a prime $p$, a subset $S$ of residues modulo $p$ is called a *sum-free multiplicative subgroup of $\mathbb{F}_p$* if:
* there is a nonzero residue $\alpha$ modulo $p$ such that $S = \{1, \alpha^1, \alpha^2, \dots\}$ (all considered mod $p$), and
* there are no $a, b, c \in S$ (not necessarily distinct) such that $a + b \equiv c \pmod p$.
Prove that for every integer $N$, there is a prime $p$ and a sum-free multiplicative subgroup $S$ of $\mathbb{F}_p$ such that $|S| \ge N$.
* there is a nonzero residue $\alpha$ modulo $p$ such that $S = \{1, \alpha^1, \alpha^2, \dots\}$ (all considered mod $p$), and
* there are no $a, b, c \in S$ (not necessarily distinct) such that $a + b \equiv c \pmod p$.
Prove that for every integer $N$, there is a prime $p$ and a sum-free multiplicative subgroup $S$ of $\mathbb{F}_p$ such that $|S| \ge N$.