Solve in the natural numbers the equation
9^{x}3^{x}=y^{4}+2y^{3}+y^{2}+2y^{}
Factor both sides
3^x (3^x 1) = y(y+2)(y^2+1)
Observe 3^x has factors of 3 so either
Case 1 y is divisible by 3: y=0mod3
Case 2 y+2 is divisible by 3: y=1mod3
(y^2+1) is never divisible by 3
(3^x1) is never divisible by 3.
Case 1:
y=3^x
and
3^x1 = (y+2)(y^2+1)
A quick graph shows this system has no solutions
Case 2
y+2=3^x
and
3^x1=y(y^2+1)
A quick graph shows these two curves cross only at (1,1)
https://www.desmos.com/calculator/orsd08h2ph
You could prove the above cases with higher math than I feel like doing. The graphs show pretty clearly.

Posted by Jer
on 20210212 11:10:29 