1993 Publications

On integers, without large prime factors, in arithmetic progressions I, Acta Mathematica, 170 (1993), 255-273.

We give the expected asymptotic for the number of $$y$$-smooth integers up to $$x$$, in the arithmetic progression $$a \pmod q$$ provided $$q\leq y^\epsilon$$.

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Solution to a problem of Bombieri, Atti della Accademia Nazionale dei Lincei, 9 (1993), 181-183.

If $$a_1,a_2,\ldots$$ is a sequence of non-negative real numbers satisfying $$ma_m + \sum_{1\leq i\leq m-1} a_ia_{m-i}=2m+O(1)$$, then either $$a_m=1+o(1)$$ or $$a_m=1-(-1)^m+o(1)$$.

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Smoothing "smooth" numbers (with John Friedlander) Philosophical Transactions of the Royal Society, 345 (1993), 339-348.

We show that the $$y$$-smooth integers are equidistributed in intervals of length $$\sqrt{x}y^{2+o(1)}$$ around $$x$$. In 2015 this has been improved by Matomaki and Radziwill to intervals of length $$A\sqrt{x}$$ provided $$A\to \infty$$ as $$x\to \infty$$

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On integers, without large prime factors, in arithmetic progressions II, Philosophical Transactions of the Royal Society, 345 (1993), 349-362.

We give the expected asymptotic for the number of $$y$$-smooth integers up to $$x$$, in the arithmetic progression $$a \pmod q$$ provided $$q\leq y^{1-\epsilon}$$ and $$q=x^{o(1)}$$.

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An upper bound in Goldbach's problem (with Jean-Marc Deshouillers, Wladyslaw Narkiewicz and Carl Pomerance) Mathematics of Computation, 61 (1993), 209-213.

We show that 210 is the largest even integer $$2n$$ for which $$2n-p$$ is prime for every prime $$p\in [n/2,n-2]$$.

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The Kummer-Wieferich-Skula criteria for the first case of Fermat's Last Theorem, Advances in Number Theory , (ed. F. Q. Gouvea and N. Yui), New York: Oxford U. Press, 1993, 479-498.

We discuss the consequences of such criteria for the first case of Fermat's Last Theorem

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Paulo Ribenboim on his retirement, Advances in Number Theory , (ed. F. Q. Gouvea and N. Yui), New York, Oxford U. Press, 1993, 469-478.

We review Ribenboim's life and career

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Fermat's Last Theorem, a Theorem at Last (with Keith Devlin and Fernando Gouvea) Focus, 13 (1993), 3-4.

A journalistic article on Wiles' work

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The number's up for maths' greatest riddle (with Ian Katz) The Guardian, 24th June 1993, section 2, pages 1-3.

A newspaper article, written the day after Wiles' famous announcement

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