Mathematician who reshaped number theory wins prestigious Abel prize

Number theorist Gerd Faltings has won this year’s Abel Prize, one of the most prestigious awards in mathematics, the Norwegian Academy of Science and Letters in Oslo announced on 19 March.

Faltings, who is at the Max Planck Institute for Mathematics in Bonn, Germany, was awarded the prize for work proving central results in the theory of algebraic equations linking whole numbers together. The award highlights Faltings’s 1983 work on Diophantine equations, which are equations that involve the sums and powers of unknown numbers and for which the solutions have to be rational — meaning that it must be possible to write the solution as either a fraction of two whole numbers or as an integer.

His proof confirmed a conjecture proposed in 1922 by mathematician Louis Mordell, which said that, except in special cases, such equations can have at most a finite set of solutions².

“This made a big splash in the mathematics community,” says Helge Holden, a mathematician at the Norwegian University of Science and Technology in Trondheim, who chairs the Abel Prize Committee. One of Faltings’s colleagues described his proof of Mordell’s conjecture as “one of the great moments in mathematics” when commenting on Faltings being bestowed a Fields Medal in 1986 — another prestigious award for mathematicians.

The Abel Prize, now in its 24th year, is modelled after the Nobel Prizes and comes with an award of 7.5 million Norwegian krone (around US$786,000). “It’s a nice sign of appreciation to get this prize,” Faltings says.

Irrational numbers

The Diophantine equations that Faltings studied include an example that most children learn about in school — the Pythagorean theorem, or x² + y² = z². Although the solution for the length z of the hypotenuse of a right-angled triangle is often an irrational number — such as the square root of two, the solution for a triangle with two sides that are one unit long — there are cases in which all three numbers satisfying the equation are integers. For example, 3² + 4² = 5². In fact, there are infinitely many such solutions.

The same is not true for powers n higher than 2, however. The result that made Faltings famous is that, except in some special cases, equations that involve higher powers and products of the unknown variables, such as x³y + y³z + z³x = 0, can never have an infinite number of rational solutions. (Perhaps the most celebrated mathematical result of the past 40 years was mathematician Andrew Wiles’s proof of ‘Fermat’s last theorem’, which says that for a special type of Diophantine equation, xⁿ + yⁿ = zⁿ, there are no rational solutions at all if n is greater than 2.)

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