I apologise for all these questions of Fermat's Last Theorem, but I am fascinated by the topic, even if I cannot understand all of it.
I must admit that I am not well versed in the language of modular forms or elliptic equations, but they seem quite complicated to me.
However, while reading Simon Singh's book "Fermat's Last Theorem", I found one particular bit very interesting.
I immediately thought that there was an easier way to prove Fermat's Last Theorem. It was to do with Falting's Theorem and the geometrical representations of equations like $x^n + y^n = 1$.
I quote: "Faltings was able to prove that, because these shapes always have more than one hole, the associated Fermat equation could only have a finite number of whole number solutions."
Surely, now all that is needed is to prove that a Fermat equation has infinite solutions.
Suppose we take the original equation: $$A^3 + B^3 = C^3$$ Surely we can find infinite solutions to this by doubling $A, B, C$ $$(2A)^3 + (2B)^3 = (2C)^3$$ $$8A^3 + 8B^3 = 8C^3$$ $$A^3 + B^3 = C^3$$ Surely this means there are infinitely many solutions to these equations. But now we have a contradiction, so therefore our original assumption, that $A^3 + B^3 = C^3$ has solutions, is false.
Who can point out my error as this seems a very simple step to take from Falting's to Fermat's. And surely that step wouldn't have taken years to take, especially for Andrew Wiles.