In this paper, the author proposes and proves a conjecture to be called the Pythagorean common prime factor conjecture, This conjecture states that if A^2 + B^2 = C^2, where A, B and C are positive integers, then A, B and C may have a common prime factor. The approach used in this paper is exactly the same as the approach used in proving the Beal conjecture (viXra:2012.0120 & viXra:2104.0098). To prove the Pythagorean common prime factor conjecture, one will let r, s and t be prime factors of A, B and C, respectively, such that A = Dr, B = Es, and C = Ft, where A, B and are positive integers. Then, the equation A^2 + B^2 = C^2, becomes (D^2)(r^2) + (E^2)(s^2) = (F^2)(t^2). The proof would be complete after proving that r^2 = t^2 and s^2 = t^2, which would imply that r = s = t. The proofs of the above equalities would also involve showing that the ratio, (r^2)/(t^2) =1 and the ratio (s^2)/(t^2) =1. Of the two numerical examples, 3^2 + 4^2 = 5^2 and 6^2 + 8^2 = 10^2, of the Pythagorean equation, the three terms of the first equation have no common prime factor; but the terms of the second equation have the common prime factor, 2. Perhaps, if there had been a Pythagorean common prime factor conjecture and its proof 24 years ago, Beal conjecture would have been proved 23 years ago. The main principle for obtaining relationships between the prime factors on the left side of the equation and the prime factor on the right side of the equation is that the power of each prime factor on the left side of the equation equals the same power of the prime factor on the right side of the equation. High school students can learn and prove this conjecture for a bonus question on a final class exam.