### 1995 Publications

#### On the number of solutions of the equation $$\sum_{i=1}^s x_i/d_i \pmod 1$$, and of diagonal equations in finite fields (with Shuguang Li and Sun Qi) Sichuan Daxue Xuebao, 32 (1995), 243-248.

We obtain sharp lower bounds for the number of solutions to diagonal equations in finite fields

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#### The number of fugitive primes (appendix to "Rational torsion of prime order in elliptic curves over number fields," by Sheldon Kamienny and Barry Mazur ) Astérisque, 228 (1995), 81-100

We show that there are few exceptional primes to the construction in the article of Kamienny and Mazur

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#### Harold Cramér and the distribution of prime numbers, Scandanavian Actuarial J., 1 (1995), 12- 28.

We discuss Cramer's contributions to analytic number theory in a modern context, in particular showing that if one believes Cramer's heuristic that there are infinitely many pairs of consecutive primes $$p, q$$ for which $$q-p\geq \{ 1+o(1)\} (\log p)^2$$, then one should believe that there are infinitely many pairs of consecutive primes $$p, q$$ for which $$q-p\geq \{ 2e^{-\gamma}+o(1)\} (\log p)^2$$.

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#### On the equations $$z^m=F(x,y)$$ and $$Ax^p+By^q =Cz^r$$ (with Henri Darmon) Bulletin on the London Mathematical Society, 27 (1995), .

We indicate when the Diophantine equations in the title have finitely many solutions. For example if $$1/p+1/q+1/r<1$$ then $$Ax^p+By^q =Cz^r$$ has finitely many solutions

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#### On a problem of Hering concerning orthogonal covers of $$K_n$$ (with H.-D. Gronau and R.C. Mullin) Journal of Combinatorial Theory, Series A, 72 (1995), 345-350.

It is shown that the complete digraph of type $$k$$ and order $$n$$ has a Hering configuration if and only if $$n\equiv 1 \pmod k$$, provided $$n$$ is sufficiently large.

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#### On the number of solutions to the generalized Fermat equation, Number Theory, Proceedings of CNTA IV (K. Dilcher, ed.) CMS Conference Proceedings 15 (1995), 197-207.

We look at how having three solutions to $$ax^p+by^p=cz^p$$ for given $$a,b,c$$, implies that there are non-diagonal solutions to $$r^p+s^p+t^p=u^p+v^p+w^p$$, and we expect that there are only finitely many solutions to this latter equation

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#### Unexpected irregularities in the distribution of prime numbers, Proceedings of the International Congress of Mathematicians, (Zürich, 1994) 1 (1995), 388--399.

We survey the surprising results that the number of primes in intervals and in arithmetic progressions, can be out from the expected number by a constant factor, stemming from Maier's matrix method

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#### Review of The World's Most Famous Math Problem'' by Marilyn vos Savant (with Nigel Boston) American Mathematical Monthly, 102 (1995), 470-473.

A review of this rather silly book

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