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F(r) = (p^{rp} - 1) (p - 1)/(p^r - 1) (p^p - 1)

Source: 2016 Saudi Arabia Pre-TST Level 4 1.4

September 13, 2020

number theorycoprimedividesdivisible

Problem Statement

Let

p

be a given prime. For each prime

r

, we defind the function as following

F(r) =\frac{(p^{rp} - 1) (p - 1)}{(p^r - 1) (p^p - 1)}

. 1. Show that

F(r)

is a positive integer for any prime

r \ne p

. 2. Show that

F(r)

and

F(s)

are coprime for any primes

r

and

s

such that

r \ne p, s \ne p

and

r \ne s

. 3. Fix a prime

r \ne p

. Show that there is a prime divisor

q

of

F(r)

such that

p| q - 1

but

p^2 \nmid q - 1

.

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