Nils Bruin
Department of Mathematics
Simon Fraser University
mail address:
Nils Bruin
Department of Mathematics
Simon Fraser University
Burnaby, BC
CANADA V5A 1S6

(phone not operational)
e: nbruin@sfu.ca
pgp: public key

office: SC K 10507

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Research subjects

Most of my mathematical work has been in number theory and especially in rational points on hyperelliptic curves. Should you have a curve you want to know the rational points on, I'd be happy to take a look at it (I don't promise any results, though!)

ABC-conjecture

In the research for my master's degree, I examined generalisations of the ABC-conjecture (also called the Masser-Oesterlee conjecture) towards more variables. The thesis itself contains a detailed description of Elkies' proof that the ABC-conjecture implies Mordell's conjecture (Faltings' theorem), a generalisation of the construction of n-examples from ABC-examples by Browkin and Brzezinski (giving more ways of doing this) and a list of small, fairly good 4-examples not coming from ABC-examples.

Generalised Fermat equations

In my PhD-thesis, I studied special cases of the diophantine equation xr+ys=zt. There is this remarkable theorem by Darmon and Granville

Theorem (Darmon, Granville): Let r,s,t be positive integers satisfying 1/r+1/s+1/t<1. Then the equation xr+ys=zt has only finitely solutions in pairwise coprime integers x,y,z.

The ABC-conjecture even suggests that all such solutions taken together (i.e., for varying r,s,t) should still form a finite set. There are some highly non-trivial solutions, though. The following list shows all known solutions. The small ones have been known for a long time, while the bigger ones were found by Beukers and Zagier.

1r+23 = 32 (r>6)
132+73 = 29
27+173 = 712
25+72 = 34
35+114 = 1222
177+762713 = 210639282
14143+22134592 = 657
338+15490342 = 156133
438+962223 = 300429072
92623+153122832 = 1137

Since the ABC-conjecture suggests that the complete list is finite, one may wonder if the known solutions are actually all solutions. This would imply

Conjecture (Tijdeman-Zagier, Beale Prize Problem): Let r,s,t,x,y,z be positive integers with r,s,t>2 and xr+ys=zt. Then x,y,z have a common prime factor.

In my thesis Chabauty methods and covering techniques applied to generalised Fermat equations I show a partial result for this.

Theorem: For (r,s,t)=(2,4,6), (2,6,4), (4,6,2), (2,3,8), (2,8,3), (2,4,5), (2,5,4), all integer solutions to xr+ys=zt with coprime x,y,z can be obtained from the solutions in the list above by inserting minus signs if necessary.

The employed methods first show that such solutions correspond to rational points on algebraic curves of genus > 1. Then the rational points on those curves are obtained by Chabauty methods and covering techniques.

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