

DIOPHANTINE EQUATIONS DUE TO SMARANDACHE1) 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. 5969. [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 References: [1] Ibstedt, H., Surphing on the Ocean of Numbers  A Few Smarandache Notions and Similar Topics, Erhus University Press, Vail, 1997, pp. 5969. [2] Smarandache, F., Only Problems, not Solutions!, Xiquan Publ. Hse., Phoenix, fourth edition, 1994, unsolved problem #11. 