Problems & Puzzles:
Problems
Problem 11. Here you are asked to find
primes of the following form
(p^{r}  1)/(p^{d}
1), expressible in more than one way.
 Batman asks if 31=(2^{5}
 1)/(2 1)=(5^{3} 1)/(51) is the only
primes which is expressible in more than one way
in the form (p^{r}  1)/(p^{d}
1) where p is prime also, r=>3, and d=>1.
No other has been found below 10^{10}.
 If the condition that p be
prime is relaxed the problem goes to Goormaghtigh
and we have the only known solution :
8191=(2^{13}1)/(21)=(90^{3}1)/(901).
In both cases : can you find
other examples?
(Ref. 2, p. 81, B25)
(See also Puzzle 61)
