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The main dish of IranNT2023

Source: Iran MO 3rd round 2023 NT exam , P2

August 17, 2023
number theoryprime numbers

Problem Statement

Let NN be the number of ordered pairs (x,y)(x,y) st 1≤x,y≤p(p−1)1 \leq x,y \leq p(p-1) and : xy≡yx≡1(modp)x^{y} \equiv y^{x} \equiv 1 \pmod{p} where pp is a fixed prime number. Show that : (ϕ(p−1)d(p−1))2≤N≤((p−1)d(p−1))2(\phi {(p-1)}d(p-1))^2 \leq N \leq ((p-1)d(p-1))^2 where d(n)d(n) is the number of divisors of nn
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