nLab abc conjecture

Idea

The abc conjecture (or ABC conjecture) is a number theoretic conjecture due to (Oesterlé-Masser 1985), which says that there are only finitely many integer solutions to the equation

$$a + b = c \;\;\; for \;\;\; a,b,c \geq 1 \in \mathbb{N}$$

(or instead $a+b+c= 0$) if one requires the integer numbers $a,b,c$ to have no common factor as well as having “joint power” greater than a given bound.

Here the power of $a,b,c$ is

$$P(a,b,c) \coloneqq \frac{log|c|}{log(rad(a\cdot b\cdot c))} \,,$$

where the radical $rad(n)$ of an integer $n$ is the product of all its distinct prime factors.

The precise form of the conjecture is:

Conjecture

(abc conjecture)

For any number $\epsilon\gt 0$ there are only finitely many positive relatively prime (coprime) integer solutions $(a,b,c)$ to the equation $a + b = c$ with power $P(a,b,c)\geq 1+\epsilon$.

According to (Mazur):

The beauty of such a Conjecture is that it captures the intuitive sense that triples of numbers which satisfy a linear relation, and which are divisible by high perfect powers, are rare; the precision of the Conjecture goads one to investigate this rarity quantitatively. Its very statement makes an attractive appeal to perform a range of numerical experiments that would test the empirical waters. On a theoretical level, it is enlightening to understand its relationship to the constellation of standard arithmetic theorems, conjectures, questions, etc., and we shall give some indications of this below.

Mason’s Theorem

According to Lang, one important antecedent of the abc conjecture is a simple but at the time unexpected relation for the function field case, published in 1984. Consider polynomials $f \in k[t]$ over an algebraically closed field $k$ of characteristic $0$, and define $n_0(f)$ to be the number of distinct roots of $f$, counted without regard to multiplicity.

Theorem (Mason)

Let $a, b, c \in k[t]$ be relatively prime polynomials, not all constant, such that $a + b = c$. Then $\max \{deg(a), deg(b), deg(c)\} \leq n_0(a b c) - 1$.

Proof

Let $f = a/c$, $g = b/c$, so that $f + g = 1$. Taking the derivative, we obtain

$$\frac{f'}{f} f + \frac{g'}{g} g = 0$$

whence

$$b/a = g/f = -\frac{f'/f}{g'/g}.$$

Put

$$a(t) = c_1 \prod (t - \alpha_i)^{m_i}, \qquad b(t) = c_2 \prod (t - \beta_j)^{n_j}, \qquad c(t) = c_3 \prod (t - \gamma_k)^{p_k}.$$

Then

$$\frac{b}{a} = -\frac{f'/f}{g'/g} = -\frac{\sum \frac{m_i}{t - \alpha_i} - \sum \frac{p_k}{t - \gamma_k}}{\sum \frac{n_j}{t - \beta_j} - \sum \frac{p_k}{t - \gamma_k}} .$$

A common denominator for $f'/f$ and $g'/g$ is given by

$$N_0 = \prod (t - \alpha_i) \prod (t - \beta_j) \prod (t - \gamma_k)$$

whose degree is $n_0(a b c)$. We then have

$$\frac{b}{a} = -\frac{N_0 f'/f}{N_0 g'/g}$$

where the numerator and denominator on the right are polynomials. However, since $b$ and $a$ are relatively prime, the fraction $b/a$ is already in lowest terms. From this we conclude that $deg(b) \leq deg(N_0 f'/f) \leq n_0(a b c) - 1$, and similarly $deg(a) \leq deg(N_0 g'/g) \leq n_0(a b c) - 1$, which completes the proof.

Corollary (FLT for polynomials)

Assume $x, y, z \in k[t]$ are relatively prime polynomials, not all constant, and suppose $x^n + y^n = z^n$. Then $n \leq 2$.

Proof

From Mason’s theorem, we conclude $n deg(x) = deg(x^n) \leq deg(x) + deg(y) + deg(z) - 1$, and similarly upon replacing $x$ by $y$ and $z$ on the left. Adding the results, we have

$$n(deg(x) + deg(y) + deg(z)) \leq 3(deg(x) + deg(y) + deg(z)) - 3$$

which is impossible if $n \geq 3$.

Guided by analogies between the ring of integers and the ring of polynomials in one variable, and building on insights of Mason, Frey, Szpiro, and others, Masser and Oesterlé were led to formulate the abc conjecture for integers as follows. Again define $N_0(m)$ for $m$ a non-zero integer to be the number of distinct primes dividing $m$.

• Conjecture: For all $\epsilon \lt 0$ there exists $C(\epsilon) \lt 0$ such that for relatively prime integers $a, b, c$ satisfying $a + b = c$, we have
  $$\max({|a|}, {|b|}, {|c|}) \leq C(\epsilon)N_0(a b c)^{1+\epsilon}.$$

Of course, this differs from the polynomial case because of the presence of $1+ \epsilon$ in the exponent, but this is a necessary evil. For example, for any $C \gt 0$, we can find relatively prime $a$, $b$, $c$ with $a + b = c$ and ${|a|} \gt C N_0(a b c)$: take $a = 3^{2^n}$, $b = -1$, and observe by repeated application of $x^2 - y^2 = (x-y)(x+y)$ that $c = a + b$ is of the form $2^n d$ for some integer $d$. Taking $n$ sufficiently large, we can easily derive the claimed inequality.

Relation to other statements

The abc conjecture implies the Mordell conjecture (Elkies).

It is equivalent to the general form of Szpiro's conjecture.

• Vojta's conjecture

References

General

The abc conjecture was stated in

• Joseph Oesterlé, David Masser (1985)

Mason’s theorem was presented in

• R. C. Mason, Equations over function fields. In Number Theory, Proceedings of the Noordwijkerhout, Springer Lecture Notes 1068 (1984), 149-157.

Material on Mason’s theorem and its relation to the abc conjecture was taken from

• Serge Lang, Algebra ($3^{rd}$ Edition), Addison-Wesley (1993), 194-196.

The relation to the Mordell conjecture is discussed in

• Noam Elkies?, ABC conjecture implies Mordell, Int. Math. Research Notices 7 (1991) 99-109

The relation to Szpiro's conjecture is discussed in

• Matt Baker (notes taken by William Stein), Elliptic curves, the ABC conjecture, and points of small canonical height (pdf)

See also

• Barry Mazur, Questions about number, pdf scan

• PolyMath, ABC conjecture

• Abderrahmane Nitaj, The ABC Conjecture Homepage

• Wikipedia, abc conjecture

Alleged proof via IUT

Shinichi Mochizuki anounced the proof which the mathematical community perceives as a serious but unchecked claim. See the references at inter-universal Teichmüller theory.

Comments on the proof are at

• Mochizuki's proof of abc.

• MathOverflow, Philosophy behind Mochizuki’s work on the ABC conjecture

A popular account of the problem of the math community checking the proof is in

• Caroline Chen, The Paradox of the Proof (web)

Another alleged proof

Kirti Joshi has put out an alleged proof of the abc conjecture on the arXiv:

• Kirti Joshi, Construction of Arithmetic Teichmuller Spaces IV: Proof of the abc-conjecture (arXiv:2403.10430)

According to Peter Scholze, there is a mistake in Joshi’s proof at Proposition 6.10.7:

• Peter Scholze, Global character of ABC/Szpiro inequalities, MathOverflow (version: 2024-03-29).