Unsolved problems in mathematics: For every ε > 0, does there exist a K>0 such that for every triple of coprime positive integers a+b=c, with product d of their distinct prime factors, |a|+|b|+|c| < Kd^1+ε?

The abc conjecture in number theory was first proposed by Joseph Oesterlé and David Masser in 1985. It is stated in terms of simple properties of three integers, one of which is the sum of the other two. Although there is no obvious attack on the problem, it has already become well known for the number of interesting consequences it entails. Image File history File links Question_dropshade. ... Unsolved problems in : Note: Use the unsolved tag: {{unsolved|F|X}}, where F is any field in the sciences: and X is a concise explanation with or without links. ... Coprime - Wikipedia /**/ @import /skins-1. ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... Two or more things are distinct if no two of them are the same thing. ... In number theory, the prime factors of a positive integer are the prime numbers that divide into that integer exactly, without leaving a remainder. ... In mathematics, a conjecture is a mathematical statement which appears likely to be true, but has not been formally proven to be true under the rules of mathematical logic. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... Joseph Oesterlé is a mathematician who, along with David Masser, formulated the so-called ABC conjecture in 1985. ... David W. Masser is Professor of Mathematics at the University of Basel, in Basel, Switzerland. ... 1985 (MCMLXXXV) was a common year starting on Tuesday of the Gregorian calendar. ...

Formulation

Let

a + b = c

be three coprime positive integers, and Coprime - Wikipedia /**/ @import /skins-1. ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...

rad(abc),

called the radical of abc, be the square-free product of their distinct prime factors. In other words, the product of all the unique prime factors of the three numbers, never raising a factor to a power greater than 1. In mathematics, a square-free integer is one divisible by no perfect square, except 1. ... In mathematics, a square-free integer is one divisible by no perfect square, except 1. ... Two or more things are distinct if no two of them are the same thing. ... In number theory, the prime factors of a positive integer are the prime numbers that divide into that integer exactly, without leaving a remainder. ...

The abc conjecture states that, for any ε > 0, there exists a finite K_ε such that, for all coprime positive integers a+b=c,

Some consequences

The conjecture has not been proved, but it has a large number of interesting consequences. These include both known results, and conjectures for which it gives a conditional proof. Conditional proof is a proof that takes the form of asserting a conditional, and proving that the premise or antecedent of the conditional necessarily leads to the conclusion. ...

• Roth's theorem (proved by Klaus Roth)
• Fermat's last theorem for all sufficiently large exponents (proved in general by Andrew Wiles)
• The Mordell conjecture (proved by Gerd Faltings)

• The Erdős–Woods conjecture except for a finite number of counterexamples
• The existence of infinitely many non-Wieferich primes
• The weak form of Hall's conjecture
• The set of consecutive triples of powerful numbers is finite
• The L function L(s,(−d/.)) formed with the Legendre symbol, has no Siegel zero
• P(x) has only finitely many perfect powers for integral x for P a polynomial with at least three simple zeros. ^[1]
• A generalization of Tijdeman's Theorem

While the first group of these have now been proven, the abc conjecture itself remains of interest, because of its numerous links with deep questions in number theory. (See unifying conjectures in mathematics for some comparisons.) There are two major results of Klaus Roth in mathematics which go by the name of Roths theorem: The Thue-Siegel-Roth theorem in Diophantine approximation, which concerns the rarity to which an irrational algebraic number can be approximated by a rational number; and Roths theorem in arithmetic... Klaus Friedrich Roth (Roth is pronounced ROW-th) (29 October 1925) is a British mathematician known for work on diophantine approximation, the large sieve, and irregularities of distribution. ... Pierre de Fermat Problem II.8 in the Arithmetica of Diophantus, annotated with Fermats comment which became Fermats Last Theorem (edition of 1670). ... Andrew Wiles should not be confused with André Weil, another famous mathematician who, like Wiles, did important work in the area of elliptic curves. ... In number theory, the Mordell conjecture stated a basic result regarding the rational number solutions to Diophantine equations. ... Gerd Faltings (born 28 July 1954) is a German mathematician known for his work in arithmetic algebraic geometry. ... Consider a sequence of consecutive positive integers . ... In mathematics, a Wieferich prime is prime number p such that p² divides 2p − 1 − 1; compare this with Fermats little theorem, which states that every prime p divides 2p − 1 − 1. ... In mathematics, Marshall Halls conjecture is an open question, as of 2006, on the differences between perfect squares and perfect cubes. ... A powerful number is a positive integer m that for every prime number p dividing m, p2 also divides m. ... In number theory, a Dirichlet character is a function &#967; from the positive integers to the complex numbers which has the following properties: There exists a positive integer k such that &#967;(n) = &#967;(n + k) for all n. ... The Legendre symbol is used by mathematicians in the area of number theory, particularly in the fields of factorization and quadratic residues. ... In mathematics, more specifically in the field of analytic number theory, a Siegel zero, named after Carl Ludwig Siegel, is a type of potential counterexample to the Grand Riemann hypothesis, on the zeroes of Dirichlet L-function. ... The integers are commonly denoted by the above symbol. ... In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ... In number theory, Tijdemans theorem states that there are at most a finite number of consecutive powers. ...

Refined forms

A more precise conjecture proposed in 1996 by Alan Baker states that in the inequality, one can replace rad(abc) by Alan Baker (born on August 19, 1939) is an English mathematician. ... The feasible regions of linear programming are defined by a set of inequalities. ...

ε^−ωrad(abc),

where ω is the total number of distinct primes dividing a, b and c. A related conjecture of Andrew Granville states that on the RHS we could also put He used to be a mathematics professor in University of Georgia and was a section speaker in the 1994 International Congress of Mathematicians, together with Dr. Carl Pomerance from UGA. Categories: Mathematician stubs | Academic biography stubs ... In mathematics, LHS is informal shorthand for the left-hand side of an equation. ...

O(rad(abc) Θ(rad(abc))

where Θ(n) is the number of integers up to n divisible only by primes dividing n.

Partial results

1986, C.L. Stewart and R. Tijdeman:

1991, C.L. Stewart and Kunrui Yu:

1996, C.L. Stewart and Kunrui Yu:

where K₁ is an absolute constant, and K₂ and K₃ are positive effectively computable constants in terms of ε.

See also

• Greatest common divisor (gcd)

In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers. ...

References

1. ^ http://www.math.uu.nl/people/beukers/ABCpresentation.pdf

External links

• Weisstein, Eric W., abc Conjecture at MathWorld.
• Abderrahmane Nitaj's ABC conjecture home page
• http://www.math.columbia.edu/~goldfeld/ABC-Conjecture.pdf
• The amazing ABC conjecture
• ABC@Home Boinc Project