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如何证明 $y^2+1=2 x^4$的正整数解为 $(1,1),(239,13)$ 
people.math.wisc.edu/~ellenberg/MCAV.pdf
Another interesting case arises from the elliptic curve
\[
y^2+1=2 x^4
\]
whose integral points are related to the problem of expressing $\pi$ as a sum of rational arctangents [26,§A.12] In particular, the point $(13,239)$ corresponds to Machin's formula
\[
\pi / 4=4 \arctan (1 / 5)-\arctan (1 / 239)
\]
有很大的解$(239,13)$是因为存在一个weight-2 cuspform in level 1024 whose mod-5 Galois representation is reducible |
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