We show how various well-known arithmetic conjectures apply to known criteria for Fermat's Last Theorem

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We study whether the usual interpolation equations for Bernoulli polynomials uniquely define them mod \( p\).

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We show that for every sufficiently large \( n\) there exists an \( m\) such that \( \binom nm \) cannot be written as the sum of three squares

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We show that Kummer's conjectured asymptotic for the first factor of the class number of a cyclotomic field is untrue assuming the Elliott-Halberstam conjecture and a quantitative twin prime conjecture

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We see how some standard conjectures in Analytic Number Theory affect the usual criteria for Fermat's Last Theorem

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Let \( k(n)\) denote the number of times we have to apply the Euler \( \phi\)-function to \( n\) to obtain 1. Assuming the Elliott-Halberstam conjecture we show that \( k(n)\) has normal order \( \alpha \log n\) for some constant \( \alpha>0 \).

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We find explicit bounds for the coefficients of a divisor \( g\) of a given polynomial \( f\) in terms of the coefficients of \( f\) and the degree of \( g\). The best possible such result was later found by Boyd.

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We find infinitely many pairs of coprime integers \( a,q\) such that the least prime \( \equiv a \pmod q\) is unusually large.

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