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### Bounding the Coefficients of a Divisor of a Given Polynomial.

Monatshefte für Mathematik

### The lattice points of an n-dimensional tetrahedron. (Summary).

Aequationes mathematicae

### On Krasner's Criteria for the First Case of Fermat's Last Theorem.

Manuscripta mathematica

### The lattice points of an n-dimensional tetrahedron.

Aequationes mathematicae

Acta Arithmetica

### Recenze pořadu Horizon stanice BBC „Fermatova poslední věta‟

Pokroky matematiky, fyziky a astronomie

Acta Arithmetica

### Pretentiousness in analytic number theory

Journal de Théorie des Nombres de Bordeaux

In this report, prepared specially for the program of the , we describe how, in joint work with K. Soundararajan and Antal Balog, we have developed the notion of “pretentiousness” to help us better understand several key questions in analytic number theory.

### Solution to a problem of Bombieri

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We solve a problem of Bombieri, stated in connection with the «prime number theorem» for function fields.

Acta Arithmetica

Integers

### Cycle lengths in a permutation are typically Poisson.

The Electronic Journal of Combinatorics [electronic only]

### Zeros of Fekete polynomials

Annales de l'institut Fourier

For $p$ an odd prime, we show that the Fekete polynomial ${f}_{p}\left(t\right)={\sum }_{a=0}^{p-1}\left(\frac{a}{p}\right){t}^{a}$ has $\sim {\kappa }_{0}p$ zeros on the unit circle, where $0.500813>{\kappa }_{0}>0.500668$. Here ${\kappa }_{0}-1/2$ is the probability that the function $1/x+1/\left(1-x\right)+{\sum }_{n\in ℤ:\phantom{\rule{4pt}{0ex}}n\ne 0,1}{\delta }_{n}/\left(x-n\right)$ has a zero in $\right]0,1\left[$, where each ${\delta }_{n}$ is $±1$ with y $1/2$. In fact ${f}_{p}\left(t\right)$ has absolute value $\sqrt{p}$ at each primitive $p$th root of unity, and we show that if $|{f}_{p}\left(e\left(2i\pi \left(K+\tau \right)/p\right)\right)|<ϵ\sqrt{p}$ for some $\tau \in \right]0,1\left[$ then there is a zero of $f$ close to this arc.

### Ranks of quadratic twists of elliptic curves

Publications mathématiques de Besançon

We report on a large-scale project to investigate the ranks of elliptic curves in a quadratic twist family, focussing on the congruent number curve. Our methods to exclude candidate curves include 2-Selmer, 4-Selmer, and 8-Selmer tests, the use of the Guinand-Weil explicit formula, and even 3-descent in a couple of cases. We find that rank 6 quadratic twists are reasonably common (though still quite difficult to find), while rank 7 twists seem much more rare. We also describe our inability to find...

### Borwein and Bradley's Apéry-like formulae for $\zeta \left(4n+3\right)$.

Experimental Mathematics

### Product of integers in an interval, modulo squares.

The Electronic Journal of Combinatorics [electronic only]

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