Let \( f\) be a primitive positive integral binary quadratic form of discriminant \( -D\), and let \( r_f(n)\) be the number of representations of \( n\) by \( f\) up to automorphisms of \( f\) . In this article, we give estimates and asymptotics for the quantity
\( \sum_{n\leq x} r_f(n)^\beta\) for all \( \beta\geq 0\) and uniformly in \( D=o(x)\). As a consequence, we get more-precise estimates for the number of integers which can be written as the sum of two powerful numbers.

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Aurefeuillian factorization is a technique to partially factor numbers of the form \(b^n\pm 1\) by replacing the \( b^n\) by a polynomial in such a way that the resulting polynomial can be factored. For example, taking \( y=2^k\) in \( (2y^2)^2+1=(2y^2-2y+1)(2y^2+2y+1)\) we can partially factor \( 2^{4k+2}+1\). We show here that Schinzel determined all such identities, and then attack a generalization of the problem using Faltings' Theorem.

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This is a survey of what was then known about "prime number races". Subsequently two of my students from that time, Youness Lamzouri and Daniel Fiorilli, have proved several extraordinary results (some with collaborators) greatly extending this theory.

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We prove asymptotic formulas for residue races in terms of Dedekind sums

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In this article, distributed at the Madrid ICM, I discuss various recent developments on the theory of prime numbers

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In Spanish: La Gaceta de la RSME 12 (2009), 547-556.

The set of cycle lengths of almost all permutations in \( S_n\) are "Poisson distributed": we show that this remains true even when we restrict the number of cycles in the permutation. The formulas we develop allow us to also show that almost all permutations with a given number of cycles have a certain "normal order" (in the spirit of the Erdos-Turan theorem). Our results were inspired by analogous questions about the size of the prime divisors of "typical"S integers.

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