The linear combination of two polygonal numbers is a perfect square

By the theory of Pell equation and congruence, we study the problem about the linear combination of two polygonal numbers is a perfect square. Let P-k(x) denote the x-th k-gonal number. We show that if k >= 5, 2(k - 2)n is not a perfect square, and there is a positive integer solution (Y', Z') of Y-2 - 2(k - 2)nZ(2) = (k - 4)(2) n(2) - 8(k - 2)n satisfying Y' + (k - 4)n 0 (mod 2(k - 2)n), Z' 0 (mod 2), then the Diophantine equation 1-FnP(k)(y) = z(2) has infinitely many positive integer solutions (y, z). Moreover, we give conditions about in, n such that the Diophantine equation mP(k)(x)+nP(k)(y) = z(2) has infinitely many positive integer solutions (x, y, z).