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= (5^3-1) / (5-1) (See your Problem 11) and the same for 8191 (and 7 ! )"
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?