1. ## Divisor Questions

1) Which positive integers have exactly four positive divisors?

2) Let n be a positive integer. Show that $\tau(2^n-1) \geq \tau(n)$

2. Originally Posted by Janu42
1) Which positive integers have exactly four positive divisors?

2) Let n be a positive integer. Show that $\tau(2^n-1) \geq \tau(n)$
1) Integers of the form $\,p_1p_2$ or $\,p^3$.

Because $\tau(n) = \prod(\alpha_i+1)$ for $n=p_1^{\alpha_1}\cdots$.

2) Haven't solved, except that it's obviously true for n prime.

3. For 2, note that if $d\mid n$ then $2^d-1 \mid 2^n-1$