### 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 = 2x_{1} x_{2} ... x_{k} +1

has infinitely many solutions in distinct primes y, x_{1} , x_{2} , ..., 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.