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Bosnia and Herzegovina TST 2012 Problem 4

Source:

May 20, 2012
functioninductionalgebrapolynomialmodular arithmeticnumber theory proposednumber theory

Problem Statement

Define a function f:N→Nf:\mathbb{N}\rightarrow\mathbb{N}, f(1)=p+1,f(1)=p+1, f(n+1)=f(1)⋅f(2)⋯f(n)+p,f(n+1)=f(1)\cdot f(2)\cdots f(n)+p, where pp is a prime number. Find all pp such that there exists a natural number kk such that f(k)f(k) is a perfect square.
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