Problems & Puzzles: Puzzles Puzzle 61. M(a, p) = a^p  a + 1 ( by Jean Brette) "I call (but probably I'm not the first to look these numbers) 'Generalized Mersenne Numbers (GMN)' the primes of the following form: M(a, p) = a^p  a + 1, p = prime a = 2 gives the standard Mersenne numbers. I have found only six GMN prime numbers with two different expressions:
Brette sees some relation of this kind of numbers and our Problem No. 11: "Of course we meet 31 since 6^2  5 = (5+1)^2 5 = 5^2+5+1= (5^31) / (51) (See your Problem 11) and the same for 8191 (and 7…! )" Brette asks: 1) "Is there other such GMN prime number greater than 78121?" I (C. Rivera) have made a little search around, and have found only two GMN composite small numbers: 2185= M(3, 7) = M(13, 3) My questions are: 2) Can you found other GMN
composite numbers of this type? 




