DIOPHANTINE EQUATIONS DUE TO SMARANDACHE





1) Conjecture:

   Let k > 0 be an integer.  There is only a finite number of solutions in

   integers p, q, x, y, each greater than 1, to the equation

    p    q

   x  - y  = k.



   For k = 1 this was conjectured by Cassels (1953) and proved by Tijdeman

   (1976).



References:

  [1] Ibstedt, H., "Surphing on the Ocean of Numbers - A Few Smarandache

      Notions and Similar Topics", Erhus University Press, Vail, 1997,

      pp. 59-69.

  [2] Smarandache, F., "Only Problems, not Solutions!", Xiquan Publ. Hse.,

      Phoenix, 1994, unsolved problem #20.







2) Conjecture:

   Let k >= 2 be a positive integer.  The diophantine equation

   y = 2x x ... x +1

         1 2     k

   has infinitely many solutions in distinct primes y, x , x , ..., x .



                                                        1   2        k

References:

  [1] Ibstedt, H., "Surphing on the Ocean of Numbers - A Few Smarandache

      Notions and Similar Topics", Erhus University Press, Vail, 1997,

      pp. 59-69.

  [2] Smarandache, F., "Only Problems, not Solutions!", Xiquan Publ. Hse.,

      Phoenix, fourth edition, 1994, unsolved problem #11.