MathDB

Let

p

be an arbitrary odd prime and

\sigma(n)

for

1 \le n \le p-1

denote the inverse of

n \pmod p

. Show that the number of pairs

(a,b) \in \{1,2,\cdots,p-1\}^2

with

a<b

but

\sigma(a) > \sigma(b)

is at least

\left \lfloor \left(\frac{p-1}{4}\right)^2 \right \rfloor

usjl
Note: Partial credits may be awarded if the

4

in the statement is replaced with some larger constant

Problem Statement