Problems & Puzzles: Puzzles

Puzzle 517. φ(x) + σ(x+1) = 3x Farideh Firoozbakht poses the following puzzle: Two questions about solutions of the equation φ(x) + σ(x+1) = 3x :   If q =2^p-1 is a Mersenne prime then q is a solution for the following equation.                                        φ(x) + σ(x+1) = 3x        (*) Because, φ(q) = q -1 = 2^p -2 , σ(q+1) = σ(2^p) =2^(p+1) -1 so  φ(q) + σ(q+1) = 3(2^p -1) = 3q. Q1. Is it true that, there is no other prime solution for (*) ? Q2.  Is it true that, Mersenne primes are the only solutions of (*) ?

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