Andrew Granville's Home Page

1995 Publications

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

Article

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

Article

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 \( q-p\geq \{ 2e^{-\gamma}+o(1)\} (\log p)^2\).

Article

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

Article

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.

Article

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

Article

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

Article

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

Article