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Mathematics · September 8, 2026

Number Theory and Elliptic Curves: The ABC Conjecture, Szpiro's Conjecture, and Their Relation to Cryptography

Why does an equation as simple as "a + b = c" — something even a middle schooler could write down — count as one of the biggest unsolved problems in modern number theory? We trace how the ABC conjecture and Szpiro's conjecture connect via elliptic curves.

This article is a sequel to the previous deep math of elliptic curve cryptography article. Rather than direct applications to cryptographic implementations, it covers "elliptic curves as basic science" — how elliptic curves connect to unsolved problems in number theory.

What Is the ABC Conjecture?

Suppose pairwise coprime positive integers a, b, c satisfy a + b = c. Define the radical rad(n) of an integer n as the product of all distinct prime numbers dividing n (for example, rad(72) = rad(2³·3²) = 2·3 = 6).

The ABC conjecture (formulated around 1985 by Oesterlé and Masser) states the following.

For every ε > 0, there exist only finitely many coprime triples (a, b, c) with a + b = c satisfying c > rad(abc)^(1+ε).

Intuitively, this says that "if a, b, c contain a lot of repeated prime factors (power factors such as squares or cubes), the simple equation a + b = c can almost never hold." Since rad(abc) only counts the "information content" of the prime factors contained in a, b, c, this conjecture states that there is a strong constraint between two seemingly unrelated properties: "the additive relationship" and "the multiplicative structure (prime factorization)."

Bridging to Elliptic Curves: The Frey Curve

The construction that connects the ABC conjecture to elliptic curves is the Frey curve. From pairwise coprime integers a, b, c (with a + b = c), we construct the following elliptic curve. 

E_{a,b,c} : y² = x(x - a)(x + b)

This curve's discriminant takes the form Δ = 16(abc)², so the prime-factorization information of a, b, c is directly encoded into the elliptic curve's discriminant. Meanwhile, this curve's conductor N (an invariant summarizing the primes at which the curve has "bad reduction") is known to be close in value to rad(abc). In other words, we have the following correspondence:

  • Information including the "multiplicity" of the prime factors of abc → the elliptic curve's discriminant Δ
  • Information only about the "kinds" of prime factors of abc (without multiplicity) → the elliptic curve's conductor N

Translated into the language of elliptic curves, the ABC conjecture's claim that "c can't be too large compared to rad(abc)" corresponds to the property that "the discriminant can't be too large compared to the conductor."

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Szpiro's Conjecture

The direct formulation of this "discriminant can't be too large compared to the conductor" property, stated purely in the language of elliptic curves, is Szpiro's conjecture (around 1981).

For every ε > 0, there exists a constant C_ε such that for every elliptic curve E over the rational numbers (with minimal discriminant Δ_E and conductor N_E), |Δ_E| ≤ C_ε · N_E^(6+ε) holds.

Via the Frey curve construction, it's known that the ABC conjecture and Szpiro's conjecture are essentially equivalent (a result due to Frey, Szpiro, and others). In other words, an elementary-sounding conjecture about the additive and multiplicative structure of integers (the ABC conjecture) and a conjecture about an invariant of an algebraic-geometric object — an elliptic curve (Szpiro's conjecture) — are two sides of the same claim, joined by the bridge of the Frey curve.

Relationship to Fermat's Last Theorem

The Frey curve was originally devised to attack Fermat's Last Theorem (that x^n + y^n = z^n has no positive integer solutions for n ≥ 3). The skeleton of Wiles's proof was to show that if a counterexample x^n + y^n = z^n existed, the Frey curve constructed from it would be a strange elliptic curve that is "not modular," contradicting the modularity theorem (the Taniyama–Shimura–Weil conjecture, which states that every elliptic curve over the rational numbers is modular). Note that this is an independent proof path that does not go through the ABC conjecture.

Separately, it is also known that a result resembling Fermat's Last Theorem follows asymptotically from the ABC conjecture (not as a proven theorem, but as a consequence conditional on the conjecture). Assuming the ABC conjecture, it follows (with finitely many exceptions) that for sufficiently large n, x^n + y^n = z^n has no solutions. This is a corollary obtained independently of Wiles's proof, derived as a consequence of the "stronger assumption" that the ABC conjecture holds.

Shinichi Mochizuki's Claimed Proof via IUT Theory

As for the ABC conjecture itself, in 2012 Shinichi Mochizuki (Research Institute for Mathematical Sciences, Kyoto University) announced a proof using a new framework of his own construction called inter-universal Teichmüller theory (IUT theory). This series of papers was published in 2020 in Publications of the Research Institute for Mathematical Sciences, a peer-reviewed journal published by RIMS itself. Mochizuki, along with collaborators including Go Yamashita and Ivan Fesenko, has consistently maintained the position that the IUT-theoretic proof is already complete. Kirti Joshi, who independently reviewed and worked with Mochizuki's framework, has also reported results supporting the proof in a "Final Report," conditional on certain supplementary extensions of his own.

On the other hand, Peter Scholze and Jakob Stix identified what they described as an apparently unfixable gap in the core of the proof (in the derivation of the so-called "inequality") in 2018. Assessment of that objection remains divided within the number theory community as of this writing (September 2026). The fact that the papers were published in a journal where Mochizuki himself held an editorial role has also led some to question the independence of that review process, and the situation is sometimes summarized as "a theorem in Kyoto, a conjecture everywhere else."

In short, the debate over a proof of the ABC conjecture comes down to two positions that coexist side by side: Mochizuki's side consistently maintaining that the proof is already complete, and Scholze and Stix's identification of an unresolved gap. In this article, we've tried to accurately convey the substance of Mochizuki's own claim, while also being clear that broad consensus across the number theory community has not been reached.

Summary

  • The ABC conjecture states a constraint between the additive relationship "a + b = c" and the multiplicative structure information rad(abc)
  • Via the Frey curve construction, an integer triple (a, b, c) translates into an elliptic curve's discriminant and conductor
  • Szpiro's conjecture is this "elliptic-curve version of the ABC conjecture" after translation, and the two are essentially equivalent
  • Fermat's Last Theorem was resolved by Wiles's proof via the modularity theorem (the Taniyama–Shimura–Weil conjecture), but a separate asymptotic route via the ABC conjecture also exists
  • For the ABC conjecture itself, Mochizuki's IUT-theoretic proof has been published in a peer-reviewed RIMS journal and Mochizuki's side maintains it is complete, while Scholze and Stix point to an unresolved gap — the number theory community as a whole has not reached consensus

Back to How Elliptic Curves Are Used in Cryptography

The algebraic derivation of the addition law is covered in the previous article.

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