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New Theorem Proved By Terry Tao

May 2004

Terry Tao, former Adelaide student and Australian IMO Gold Medallist in 1988 at the age of 12 (a feat which has never been equalled), has recently proved a mathematics theorem which has drawn wide attention and praise from within the mathematics community.

Number Theory is a glamour area of pure mathematics and the world's top mathematicians have worked for centuries on problems form this field. The distribution of prime numbers has attracted particular attention. The most important outstanding problem in all of mathematics is to be found here as Riemann's Hypothesis, which was first postulated over 150 years ago. Much of the work in getting where we are in resolving Riemann's Hypothesis has involved methods from analysis.

A particular property which is well known is the existence of pairs (with distance of two between them). Examples of this are 3 and 5, 5 and 7, 11 and 13, 17 and 19. Except for 3, 5 and 7 triples are not possible as one member would in turn have to be divisible by 3.

However the existence of larger arithmetic sequences with different "differences" has also attracted considerable interest. The eminent Cambridge mathematician GH Hardy spent much of his time pursuing the existence of other, longer arithmetic progressions.

Certainly many longer progressions, with larger differences can be found, such as 5, 17, 29, 41 and 53.

Hardy certainly believed, without being able to prove, that there were was no upper limit on the length of such a sequence.

Terry Tao, now at University of California at Los Angeles, still only 28 years old, and Ben Green, a former member of the UK IMO team and at the University of British Columbia in Vancouver (at the time of this work), have proved that there is indeed no upper limit. Arithmetic Progressions of any length can be found.

Interestingly, Terry and ben have applied some very original ideas, also using mathematical analysis in solving this number theory problem.

This is a ground-breaking result which has attracted considerable attention.

References include New Scientist, 08 May 2004 and this maths abstracts web site.

David Hunt & Terry Tao

Terry (right) is shown with David Hunt, leader of the Australian IMO team in Washington in 2001, while Terry was in Washington.


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