Search Results

You are looking at 1 - 2 of 2 items for

  • Author: Susil Kumar Jena x
Clear All Modify Search
Open access

Susil Kumar Jena

Abstract

In this paper, we prove that for any positive integer p, when p ≡ 1 (mod 6) or, p ≡ 3 (mod 6), the Diophantine equation: 2p A + B = C has infinitely many co-prime integral solutions A, B, C. When p = 0, this equation has only four integral solutions with (A, B, C) = (±1, 2, ±3). For other integer values of p, the problem is open.

Open access

Susil Kumar Jena

Abstract

In p. 219 of R.K. Guy's Unsolved Problems in Number Theory, 3rd edn., Springer, New York, 2004, we are asked to prove that the Diophantine equation xn + yn = n!zn has no integer solutions with n ∈ N+ and n > 2. But, contrary to this expectation, we show that for n = 3, this equation has in finitely many primitive integer solutions, i.e. the solutions satisfying the condition gcd(x, y, z) = 1.