I'm reading the article On the equation $a^p + 2^\alpha b^p + c^p = 0$ by Ribet (http://math.berkeley.edu/~ribet/Articles/acta.pdf), but I'm having trouble understanding his proof of Theorem 3. For example, I don't get why Theorem 3 is the same thing as showing that $t=5$ (3rd line on page 7 of the PDF file); also, what is the explicit contradiction which shows that there are no non-trivial primitive solutions? [Sorry if these are trivial questions, I'm just beginning learning the modular approach to Diophantine equations!]
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3$\begingroup$ One has (line 5 of Section 2) that $B=2^\alpha b^p$ in the Frey curve. The text over the page break of 6-7 says that $t=5$ exactly when $4$ does not divide $B$ (same is given at the top of page 5). Now what Ribet means by his statement is: "We now show Theorem 3, that is, $t\neq 5$ is impossible and thus there are no solutions with $2\le\alpha<p$ (and also no solutions with $\alpha=1$ where $2|abc$)." His paragraph on page 7 gives a few more details. And as stated there, the contradiction is to the lack of weight 2 modular forms of level 8. $\endgroup$– ConderCommented Apr 29, 2014 at 0:27
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$\begingroup$ Ok, that makes sense! I guess I was slightly confused by his wording. Thanks $\endgroup$– user50160Commented Apr 29, 2014 at 7:03
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$\begingroup$ @Conder: It seems this is what user50160 asked for. Would you mind formulating it as an answer? $\endgroup$– András BátkaiCommented Apr 29, 2014 at 9:30
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