Problems & Puzzles: Problems

Problem 11.- Here you are asked to find primes of the following form
r - 1)/(pd -1), expressible in more than one way.

  1. Batman asks if 31=(25 - 1)/(2 -1)=(53 -1)/(5-1) is the only primes which is expressible in more than one way in the form (pr - 1)/(pd -1) where p is prime also, r=>3, and d=>1. No other has been found below 1010.
  2. If the condition that p be prime is relaxed the problem goes to Goormaghtigh and we have the only known solution :


In both cases : can you find other examples?

(Ref. 2, p. 81, B25)

(See also Puzzle 61)

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