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We prove two supercongruences involving Almkvist-Zudilin sequences, which were originally conjectured by Z.-H. Sun (2020).
We show that for all integers and there are no non-trivial solutions of Thue equationsatisfying the additional condition .
Let be a prime and let be a number field. Let be the Galois representation given by the Galois action on the -adic Tate module of an elliptic curve over . Serre showed that the image of is open if has no complex multiplication. For an elliptic curve over whose -invariant does not appear in an exceptional finite set (which is non-explicit however), we give an explicit uniform lower bound of the size of the image of .
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