A CEILING FOR A + B = C, WITH EXPONENT 24

Write down a + b = c with three whole numbers that share no common factor. Now list the prime numbers that divide a, b or c, and multiply them together, each one only once. Call the result N.

Most of the time, N is much bigger than c. Take 2 + 3 = 5: the primes are 2, 3 and 5, so N = 30, far above 5. But sometimes the numbers are built from very few, very small primes. In 1 + 8 = 9, the only primes are 2 and 3, so N = 6 — less than c. In 1 + 80 = 81, with 80 = 2⁴ × 5 and 81 = 3⁴, N is 30, while c is 81.

Small primes, big sums?

How far can this go? Can c be enormous while N stays small? The abc conjecture of Masser and Oesterlé says it cannot: c stays below a constant times a power of N, and that power can be taken as close to 1 as you like. Several other statements about whole numbers are known to follow from it.

In a preprint posted on 3 November 2026, Christophe Soulé, of the CNRS and the Institut des hautes études scientifiques (IHÉS) in Bures-sur-Yvette, France, announces a proof of a weak form of it:

c ≤ A(ε) · N^(24+ε)

Here ε is any positive number, as small as you like, and A(ε) a constant that depends only on it. In words: c can never exceed a fixed multiple of N to the power of roughly 24. For N = 30, that ceiling is around 3 × 10³⁵ times a constant — enormous, but fixed.

The author is explicit that this is not the conjecture itself: the abc rule follows here “albeit with a wrong exponent”, as the paper puts it.

A detour through curves

The proof takes a detour. Its main result concerns curves of genus 2 — equations in two unknowns of a particular complexity. For such a curve, the paper bounds the “height” of its solutions, a measure of how large their numbers are to write down, by 12 + ε times a quantity attached to the number system the solutions live in. A more ambitious conjecture by Vojta would put 1 + ε in place of 12 + ε.

It was already known, from work by Moret-Bailly, that such a bound implies a form of abc. The strategy itself was proposed by Parshin, then Moret-Bailly. Soulé carries it out:

1. A construction due to Kodaira attaches to the genus-2 curve a whole family of curves of genus 6.
2. A quantity measured on each curve of the family, its arithmetic “self-intersection”, turns out to be a height. The core theorem puts a ceiling on it, with a factor 36 + ε.
3. To get there, the paper combines intersection theory for arithmetic surfaces, Minkowski’s theorem on successive minima, and computations on curves of genus 3 and 6 carried out “with the help of” Voisin.

The exponent 24 then falls out of one specific curve, y² + y = x⁵. Two technical steps on this curve turn the factor 12 of the genus-2 bound into the exponent: the first, which uses a map of degree 5 from the curve onto a line, multiplies it by 5/2, giving 30; the second, an estimate of a quantity called a discriminant, multiplies it by 4/5, giving 24.

Consequences, with weaker constants

According to the paper, the known consequences of abc follow from its results too, “with weaker constants”. It proves two examples:

• a weak form of Szpiro’s conjecture on elliptic curves, with a factor of 144 (1 + log 24) + ε;
• the absence of so-called Siegel zeros for a family of functions of number theory, L-functions of quadratic characters with negative discriminant — by an argument of Granville and Stark, which, the paper notes, works with an exponent other than 1 + ε. The paper also refers to two other recent preprints on that question.

Still to be checked

This is a 39-page proof by a single author, written in the advanced language of arithmetic geometry, and it has not yet been through peer review. The author thanks 26 colleagues for their help, but that does not mean they have checked it. The paper does not say whether the constant A(ε) can be computed explicitly.

If the argument holds, the sums a + b = c will have a ceiling with a fixed exponent: not yet the one the conjecture asks for, just above 1, but 24.