Andrew Granville's Home Page

1993 Publications

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

Article

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 \( a_m=1-(-1)^m+o(1)\).

Article

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 \( x\to \infty\)

Article

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 \( q=x^{o(1)}\).

Article

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 \( p\in [n/2,n-2] \).

Article

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

Article

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

Article

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

Article

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

Article