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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