Gerd Faltings has won the 2026 Abel Prize for his work on proving that Diophantine equations can have a finite set of solutions.Credit: Peter Badge/Typos1/The Abel Prize
Gerd Faltings, a number theorist at the Max Planck Institute for Mathematics in Bonn, Germany, has won the 2026 Abel Prize, one of the most prestigious awards in mathematics, the Norwegian Academy of Science and Letters announced on 19 March.
Faltings was awarded the prize for work proving central results in the theory of algebraic equations linking whole numbers together1. The prize highlights Faltings’s work in 1983 on the theory of Diophantine equations, which are equations involving sums and powers of unknown numbers for which the solutions have to be rational — meaning they can be written as a fraction of two whole numbers, or integers.
His proof confirmed a conjecture stated in 19222 by US mathematician Louis Mordell, which said that, except in special cases, such equations can have at most a finite set of solutions.
“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 Committee. Commenting on Faltings’s 1986 award of a Fields Medal — another of the greatest honours for a mathematician — a colleague described his proof of Mordell’s conjecture as “one of the great moments in mathematics”.
The Abel Prize, now in its 24th year, is modelled after the Nobel Prizes and comes with an award of 7.5 million Norwegian Kroner (US$780,000). “It’s a nice sign of appreciation to get this prize,” Faltings says.
Faltings was attracted to the field of mathematics for its ‘intellectual clarity’. Credit: Peter Badge/Typos1/The Abel Prize
Irrational numbers
The type of equations that Faltings studied includes an example that most children learn in school — the Pythagorean identity 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 √2 — there are cases where 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 unknowns — such as x3y + y3z + z3x = 0 — can never have an infinite number of rational solutions. (Perhaps the most celebrated mathematical result of the last 40 years was British mathematician Andrew Wiles’ 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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