Question 25
Denote by $ d(n) $ the number of divisors of the positive integer n. A positive integer n is called highly divisible if $ d(n) \gt d(m) $ for all positive integers m < n. Two highly divisible integers m and n with m < n are called consecutive if there exists no highly divisible integer s satisfying m < s < n.
(a) Show that there are only finitely many pairs of consecutive highly divisible integers of the form $ (a,b) $ with a |b.
(b) Show that for every prime number $p$ there exist infinitely many positive highly divisible integers $r$ such that $pr$ is also highly divisible.
(a) Show that there are only finitely many pairs of consecutive highly divisible integers of the form $ (a,b) $ with a |b.
(b) Show that for every prime number $p$ there exist infinitely many positive highly divisible integers $r$ such that $pr$ is also highly divisible.