LAUSR

Let m be a positive integer, and let p be a prime with $$p \equiv 1~(\mathrm{mod}~4).$$p≡1(mod4). Then we show that the exponential Diophantine equation $$(3pm^2-1)^x+(p(p-3)m^2+1)^y=(pm)^z$$(3pm2-1)x+(p(p-3)m2+1)y=(pm)z has only the positive integer solution $$(x, y, z)=(1, 1, 2)$$(x,y,z)=(1,1,2) under some conditions. As a corollary, we derive that the exponential Diophantine equation $$(15m^2-1)^x+(10m^2+1)^y=(5m)^z$$(15m2-1)x+(10m2+1)y=(5m)z has only the positive integer solution $$(x, y, z)=(1, 1, 2).$$(x,y,z)=(1,1,2). The proof is based on elementary methods and Baker’s method.