Show that the following are composite integers by finding a non-trivial factor of each integer:
(a) $2^{123} - 1$
(b) $2^{161051} - 1$
(c) $2^{1769} - 1$
We'll be using Proposition (4.9) which says that for positive integers $m,n$
$$ m \mid n \implies (2^m - 1) \mid (2^n - 1) $$
(a) Using $123=3 \times 41$ we have
$$ 3 \mid 123 \implies (2^{3}-1) \mid (2^{123}-1) $$
So a non-trivial factor is $2^3-1 = 7$.
(b) Using $161051 = 11^5$ we have
$$ 11 \mid 161051 \implies (2^11 - 1) \mid (2^{161051}-1)$$
So a non-trivial factor is $2^11-1 = 2047$
(c) Using $1769=29 \times 61$ we have
$$ 29 \mid 1769 \implies (2^{29}-1) \mid (2^{1769}-1) $$
So a non-trivial factor is $2^{29}-1 = 536870911$.