Problems & Puzzles: Puzzles

Puzzle 153.  Espinosa's puzzle

José Espinosa, from Santiago de Chile, posed the following problem

• Let p be a prime number greater than 3 and q a prime number.
• Let g(n)=2^(2+n(p-1))+3^(3+n(p-1))+5^(5+n(p-1))+7^(7+n(p-1))+...+ q^(q+n(p-1))
• Let f(n)=48g(n)-(n-5)(n-7)g(13)+4(n-5)(n-13)g(7)-3(n-7)(n-13)g(5)

Prove that for every integer non-negative n: f(n) is divisible by p^3.

He also sent to me a hint that I will provide to those of you who need it.

But maybe can help enough to know that Mr. Espinosa currently maintains a math site devoted to the mathematical induction.

Solution