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:

7

  = M(2, 3)         = M(3, 2)
31   = M(2, 5) = M(6, 2)
211   = M(6, 3) = M(15, 2)
241   = M(3, 5) = M(16, 2)
8191   = M(2, 13) = M (91, 2)
78121   = M(5, 7) = M(280, 2)"

 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…! )"

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)
24299971 = M(30, 5) = M(4930, 2)

 My questions are:

2) Can you found other GMN composite numbers of this type?
3) Do exist other GMN (prime or composite) numbers of this type expressible in 3 or more ways?
4) Is there other GMN prime greater than 78121 where p is not 2 in the second expression?
5) Is there any GMN number - other than 7 & 2185 - where a & p are prime numbers in both expressions?
6) Is there any GMN number - other than 7 - where a & p are transposed between the two expressions
7) Is there any special reason why, except the first GMN prime of this kind of numbers (7), all the other end in the decimal digit "one".


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