A note on the Diophantine Equation p + q + r = M2 and the Goldbach Conjectures

Abstract

Abstract. We discuss the title equation with primes p, q, r, and the connection to the binary and ternary Goldbach Conjectures. The binary conjecture states the every even integer N ≥ 4 is the sum of two primes, whereas the ternary conjecture states that every odd integer N ≥ 7 is the sum of three primes. Under the assumption that the binary conjecture is true, it is established for each and every fixed prime p ≥ 2 and also for each and every M ≥ 3, that there exist primes q, r so that the equation has at least one solution. The ternary conjecture is also verified in the case of this equation.

Three Theorems on the Goldbach Conjecture, and the Intimate Prime-Pairs

“…The gaps between consecutive primes have been of perennial interest; especially the twin primes (see [10,11,12,13,14]). It is interesting that the relation between Diophantine equation and Goldbach conjecture for a particular case of the equation (see [15,16]). Because of the plausibility of the Goldbach conjecture, it seems likely that 5 is the only odd untouchable number (see [8,17,18]).…”

Solutions of the Diophantine Equations p x + (p + 1)y + (p + 2)z = M 2 for Primes p ≥ 2 when 1 ≤ x, y, z ≤ 2

“…In (17) the number 8 = 2 3 has an odd exponent equal to 3. If N is even, then the factor (2N The prime 2 has an odd exponent equal to 1, and the factor (16N 2 + 34N + 19) is odd for all values N. Thus, the right side of (20) is not equal to a square.…”