### 1992 Publications

#### Limitations to the equi-distribution of primes III (with John Friedlander) Compositio Mathematicae, 81 (1992), 19-32.

We show that if $$a$$ is fixed then there are values of $$q\leq x/(\log x)^N$$, which are coprime to $$a$$, such that the asymptotic $$\pi(x;q,a) \sim \pi(x)/\phi(q)$$ fails to hold.

Article

#### Zaphod Beeblebrox's brain and the fifty-ninth row of Pascal's triangle, American Mathematical Monthly , 99 (1992), 318-331; (Corrigendum) 104 (1997), 848-851.

The number of odd entries in a row of Pascal's triangle is always a power of 2. These are either equally split between 1 and 3 mod 4, or are all 1 mod 4. Similarly, for every odd $$a$$, the number of entries in a given row of Pascal's trinagle that are $$\equiv a \pmod 8$$ is either 0 or a power of 2. We develop a theory to explain this.

Article, Corrigendum

#### Finding integers $$k$$ for which a given Diophantine Equation has no solution in $$k$$th powers of integers, Acta Arithmetica , 60 (1992), 203-212.

For a given polynomial $$f$$ we use local methods to find exponents $$k$$ for which there are no non-trivial integer solutions to the Diophantine equation $$f(x_1^k,\ldots, x_n^k)=0$$

Article

#### Squares in arithmetic progressions (with Enrico Bombieri and Janos Pintz) Duke Mathematical Journal,66 (1992), 165-204.

We show that there are $$\ll N^{2/3+o(1)}$$ squares in any given arithmetic progression of length $$N$$. This was later improved by Bombieri and Zannier to $$\ll N^{3/5+o(1)}$$; while the conjecture is that the maximum is $$\sqrt{8N/3}+O(1)$$, given by $$1, 25, 49,\ldots, 24N-23$$.

Article

#### Computation on the first factor of the class number of cyclotomic fields (with Gilbert Fung and Hugh C. Williams) Journal of Number Theory, 42 (1992), 297-312.

We compute the first factor of the class number of the $$p$$th cyclotomic field for each prime $$p\leq 3000$$.

Article

#### On elementary proofs of the Prime Number Theorem for arithmetic progressions, without characters, Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989), Univ. Salerno,Salerno, Italy, 1992, 157-194.

We show that Selberg's formula, by itself, leads to two possible behaviours for the prime number theorem in arithmetic progression. This allows us to deduce the behaviour of a possible Siegel zero using elementary methods.

Article

#### Relevance of the residue class to the abundance of primes (with John Friedlander) Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989), Univ. Salerno, Salerno, Italy, 1992, 95-104.

We show strong upper bounds for the average number of primes $$\equiv a \pmod q$$ as one varies over $$q$$ coprime to $$a$$. Asymptotics were attained much later by Fiorilli.

Article

#### Primality testing and Carmichael numbers, Notices of the American Mathematical Society, 39 (1992), 696-700.

A survey on Carmichael numbers just after it was proved that there are infinitely many

Article