Cameron Leigh Stewart

Contributions

He has made numerous contributions to number theory, in particular to work on the abc conjecture. In 1976 he obtained, with Alan Baker, an effective improvement to Liouville's Theorem. In 1991 he proved that the number of solutions to a Thue equation ${\displaystyle f(x,y)=h}$ is at most ${\displaystyle 2800(1+1/4ϵ\mathrm{deg}f)(\mathrm{deg}f{)}^{1+\omega (g)}}$, where ${\displaystyle ϵ}$ is a pre-determined positive real number and ${\displaystyle \omega (g)}$ is the number of distinct primes dividing a large divisor ${\displaystyle g}$ of ${\displaystyle h}$. This improves on an earlier result of Enrico Bombieri and Wolfgang M. Schmidt and is close to the best possible result. In 1995 he obtained, along with Jaap Top, the existence of infinitely many quadratic, cubic, and sextic twists of elliptic curves of large rank. In 1991 and 2001 respectively, he obtained, along with Kunrui Yu, the best unconditional estimates for the abc conjecture. In 2013, he solved an old problem of Erdős (so his Erdős number is 1) involving Lucas and Lehmer numbers. In particular, he proved that the largest prime divisor ${\displaystyle P(n)}$ of ${\displaystyle 2^n-1}$ satisfies ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}P(n)/n=\mathrm{\infty }}$.