How many solutions does the following exponential diophantine equation have in the positive integers? $$37+\frac{3(x-1)x}2+3^{x+2}=y^2$$ The solutions $\,(x,y)\,$ I have found are: $\;(1,8),\;(2,11),\;(3,17),\;(4,28)$.
Are the only solutions?
Many thanks.
P.S. The equation has been built noting that: $$27+37+0=64=8^2$$ $$81+37+3=121=11^2$$ $$243+37+9=289=17^2$$ $$729+37+18=784=28^2$$
The following can be found with the 15-digit precision of Excel.
$$ (1,8)\quad (2,11)\quad (3,17)\quad (4,28)\quad (36,1162261467)\quad (38,3486784401)\quad (40,10460353203)\quad (42,31381059609)\quad (44,94143178827)\quad (46,282429536481)\quad (48,847288609443)\quad (50,2541865828329)\quad (52,7625597484987)\quad (54,22876792454961)\quad (56,68630377364883)\quad (57,118871300538602)\quad (58,205891132094649)\quad (59,356613901615807)\quad (60,617673396283947)\quad $$ But WolframAlpha disagrees with the $x\ge36$ as seen here so, if there are other pairs, it would take software with arbitrary precision to find them.