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

x^p - y^q = 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 = 2x₁ x₂ ... x_k +1

has infinitely many solutions in distinct primes y, x₁ , x₂ , ..., x_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.