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abc conjecture

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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=cfora,b,c1 a + b = c \;\;\; for a,b,c \geq 1 \in \mathbb{N}

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

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

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

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

The precise form of the conjecture is:

Conjecture

(abc conjecture)

For any number ϵ>0\epsilon\gt 0 there are only finitely many positive relatively prime (coprime) integer solutions (a,b,c)(a,b,c) to the equation a+b=ca + b = c with power P(a,b,c)1+ϵP(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 fk[t]f \in k[t] over an algebraically closed field kk of characteristic 00, and define n 0(f)n_0(f) to be the number of distinct roots of ff, counted without regard to multiplicity.

Theorem (Mason)

Let a,b,ck[t]a, b, c \in k[t] be relatively prime polynomials, not all constant, such that a+b=ca + b = c. Then max{deg(a),deg(b),deg(c)}n 0(abc)1\max \{deg(a), deg(b), deg(c)\} \leq n_0(a b c) - 1.

Proof

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

fff+ggg=0\frac{f'}{f} f + \frac{g'}{g} g = 0

whence

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

Put

a(t)=c 1(tα i) m i,b(t)=c 2(tβ j) n j,c(t)=c 3(tγ k) p k.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

ba=f/fg/g=m itα ip ktγ kn jtβ jp ktγ k.\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/ff'/f and g/gg'/g is given by

N 0=(tα i)(tβ j)(tγ k)N_0 = \prod (t - \alpha_i) \prod (t - \beta_j) \prod (t - \gamma_k)

whose degree is n 0(abc)n_0(a b c). We then have

ba=N 0f/fN 0g/g\frac{b}{a} = -\frac{N_0 f'/f}{N_0 g'/g}

where the numerator and denominator on the right are polynomials. However, since bb and aa are relatively prime, the fraction b/ab/a is already in lowest terms. From this we conclude that deg(b)deg(N 0f/f)n 0(abc)1deg(b) \leq deg(N_0 f'/f) \leq n_0(a b c) - 1, and similarly deg(a)deg(N 0g/g)n 0(abc)1deg(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,zk[t]x, y, z \in k[t] are relatively prime polynomials, not all constant, and suppose x n+y n=z nx^n + y^n = z^n. Then n2n \leq 2.

Proof

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

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

which is impossible if n3n \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)N_0(m) for mm a non-zero integer to be the number of distinct primes dividing mm.

  • Conjecture: For all ϵ<0\epsilon \lt 0 there exists C(ϵ)<0C(\epsilon) \lt 0 such that for relatively prime integers a,b,ca, b, c satisfying a+b=ca + b = c, we have
    max(|a|,|b|,|c|)C(ϵ)N 0(abc) 1+ϵ.\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+ϵ1+ \epsilon in the exponent, but this is a necessary evil. For example, for any C>0C \gt 0, we can find relatively prime aa, bb, cc with a+b=ca + b = c and |a|>CN 0(abc){|a|} \gt C N_0(a b c): take a=3 2 na = 3^{2^n}, b=1b = -1, and observe by repeated application of x 2y 2=(xy)(x+y)x^2 - y^2 = (x-y)(x+y) that c=a+bc = a + b is of the form 2 nd2^n d for some integer dd. Taking nn 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.

References

The abc conjecture was stated in

  • Joseph Oesterlé, David Masser (1985)

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

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

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

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 rd3^{rd} Edition), Addison-Wesley (1993), 194-196.

See also

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)
Revised on May 10, 2013 16:28:36 by Urs Schreiber (82.169.65.155)