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# A Repulsion Motif in Diophantine Equations

Award: Lester R. Ford

Year of Award: 2012

Publication Information: The American Mathematical Monthly, vol. 118, no. 7, August-September 2011, pp. 584-598

Summary (From the Prizes and Awards booklet, MathFest 2012)

Beginning with $$y^2 + 2 = x^3$$, the authors entice the reader with the distinguished history of this equation along with the surprising sizes of solutions. The authors then lead the reader forward in time, effectively offering a "speed dating" tour of highlights in Diophantine equations, such as the abc conjecture, the Baker-Stark methods, the recent proof of the Catalan conjecture, and the geometry of elliptic curves. They introduce key definitions and themes of Diophantine equations in simple concrete contexts, hinting at the complexity that a fully general description would involve. The authors weave several themes throughout the article, such as the interplay of computation/conjecture/theory, or the "familiar refrain" that an effective (bounded) search may still be an impracticable one.