We show that the value of the \( (p-1)\)st Bernoulli polynomial at \( a/q\) can be given, mod \( p\), in terms of a certain linear recurrence of order \( [q/2]\), which depends only on \( a,q\) and \( p \pmod q\).

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We classify those finite simple groups whose Brauer graph has a \( p\)-block with defect 0. .

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We prove Erdos's conjecture that \( \binom {2n}n\) is not squarefree for all \( n>4\), by obtaining explicit upper bounds on exponential sums of the form \( \sum_n \Lambda(n)e(x/n) \) for \( n\sim N\ll x^{3/5}\)

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We investigate the number of subsets of the integers up to \( x\) that have certain given arithmetic properties.

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