### 1990 Publications

#### Some conjectures related to Fermat's Last Theorem Number Theory (ed. R.A. Mollin), (New York: de Gruyter), 1990, 177-192.

We show how various well-known arithmetic conjectures apply to known criteria for Fermat's Last Theorem

Article

#### Defining Bernoulli polynomials in $$\mathbb Z/p\mathbb Z$$ (A generic regularity condition) (with Herb Shank) Proceeding of the American Mathematical Society, 108 (1990), 637-640.

We study whether the usual interpolation equations for Bernoulli polynomials uniquely define them mod $$p$$.

Article

#### Representing binomial coefficients as sums of squares (with Yiliang Zhu) American Mathematical Monthly, 97 (1990), 486-493.

We show that for every sufficiently large $$n$$ there exists an $$m$$ such that $$\binom nm$$ cannot be written as the sum of three squares

Article

#### On the size of the first factor of the class number of a cyclotomic field Inventiones Mathematicae, 100 (1990), 321-338.

We show that Kummer's conjectured asymptotic for the first factor of the class number of a cyclotomic field is untrue assuming the Elliott-Halberstam conjecture and a quantitative twin prime conjecture

Article

#### Some conjectures in Analytic Number Theory and their connection with Fermat's Last Theorem Analytic Number Theory (ed. B.C. Berndt, H.G. Diamond, H. Halberstam, A. Hildebrand), (Boston: Birkhauser), 1990, 311-326.

We see how some standard conjectures in Analytic Number Theory affect the usual criteria for Fermat's Last Theorem

Article

#### On the normal behavior of the iterates of some arithmetic functions (with Paul Erdos, Carl Pomerance and Claudia Spiro) Analytic Number Theory (ed. B.C. Berndt, H.G. Diamond, H. Halberstam, A. Hildebrand), (Boston: Birkhauser), 1990, 165-204.

Let $$k(n)$$ denote the number of times we have to apply the Euler $$\phi$$-function to $$n$$ to obtain 1. Assuming the Elliott-Halberstam conjecture we show that $$k(n)$$ has normal order $$\alpha \log n$$ for some constant $$\alpha>0$$.

Article

#### Bounding the coefficients of a divisor of a given polynomial Monatshefte fur Mathematik, 109 (1990), 271-277.

We find explicit bounds for the coefficients of a divisor $$g$$ of a given polynomial $$f$$ in terms of the coefficients of $$f$$ and the degree of $$g$$. The best possible such result was later found by Boyd.

Article

#### On the least prime in certain arithmetic progressions (with Carl Pomerance) Journal of the London Mathematical Society, 41 (1990), 193-200.

We find infinitely many pairs of coprime integers $$a,q$$ such that the least prime $$\equiv a \pmod q$$ is unusually large.

Article