The abc conjecture ( also known as Oesterlé – Masser conjecture ) is a conjecture in number theory, first proposed by and as an integer analogue of the Mason – Stothers theorem for polynomials.

The conjecture is stated in terms of three positive integers, a, b and c ( whence comes the name ), which have no common factor and satisfy a + b = c. If d denotes the product of the distinct prime factors of abc, the conjecture essentially states that d cannot be much smaller than c.

The abc conjecture has already become well known for the number of interesting consequences it entails.

Many famous conjectures and theorems in number theory would follow immediately from the abc conjecture.

described the abc conjecture as " the most important unsolved problem in Diophantine analysis ".

In August 2012, Shinichi Mochizuki released a paper with a serious claim to a proof of the abc conjecture.

The abc conjecture can be expressed as follows:

The abc conjecture deals with the exceptions.

* The abc conjecture

# REDIRECT abc conjecture

* abc conjecture

Joseph Oesterlé ( born 1954 ) is a French mathematician who, along with David Masser, formulated the abc conjecture in 1985.

One of the most striking applications of the notion of radical occurs in the abc conjecture, which states that, for any ε > 0, there exists a finite K < sub > ε </ sub > such that, for all triples of coprime positive integers a, b, and c satisfying a + b = c,

This was a novel contribution to the circle of ideas around the Mordell conjecture and abc conjecture, suggesting something of large importance to the integer solutions ( affine space ) aspect of diophantine equations.

Along with Joseph Oesterlé, Masser formulated the abc conjecture in 1985.

# The number formed by its first three digits abc is a multiple of 3.

*# Multiply this number by itself plus one: abc ( abc + 1 )

It supports an unlimited number and length of staffs, polyphony, MIDI playback of written notes, chord markings, lyrics, and a number of import and export filters to many formats like MIDI, MusicXML, abc, MUP, PMX, MusiXTeX and LilyPond.

In 1972, Derrida wrote " Signature Event Context ," an essay on J. L. Austin's speech act theory ; following a critique of this text by John Searle in his 1977 essay Reiterating the Differences, Derrida wrote the same year Limited Inc abc ..., a long defense of his earlier argument.

The cartographic depictions of the southern continent in the 16th and early 17th centuries, as might be expected for a concept based on such abundant conjecture and minimal data, varied wildly from map to map ; in general, the continent shrank as potential locations were reinterpreted.

Amongst his most original contributions were: his " Conjecture II " ( still open ) on Galois cohomology ; his use of group actions on Trees ( with H. Bass ); the Borel-Serre compactification ; results on the number of points of curves over finite fields ; Galois representations in ℓ-adic cohomology and the proof that these representations have often a " large " image ; the concept of p-adic modular form ; and the Serre conjecture ( now a theorem ) on mod-p representations that made Fermat's last theorem a connected part of mainstream arithmetic geometry.

In 1960, Claude Berge formulated another conjecture about graph coloring, the strong perfect graph conjecture, originally motivated by an information-theoretic concept called the zero-error capacity of a graph introduced by Shannon.

The theorem-the improved conjecture-supersedes the primitive conjecture with the new proof-generated concept as its paramount new feature.

In 1973, he proved this conjecture false, though the concept that image textures could be modeled based on low-order statistics remained.

It is also known as the contraction clique number of G. The Hadwiger conjecture can be stated in the simple algebraic form χ ( G ) ≤ h ( G ) where χ ( G ) denotes the chromatic number of G. A related concept, the achromatic number of G, is the size of the largest clique that can be formed by contracting a family of independent sets in G ).

Historian Jones writes that, " Siegel flatly denied that Wylie's novel had influenced him in any way ," although Jones added his own conjecture that " the timing and striking similarities ... would seem to leave no doubt of Gladiator < nowiki >'</ nowiki > s role ".</ ref > The concept of a human having the proportional strength of an insect is also the basis of the Spider-Man series.

In mathematics, any number of cases supporting a conjecture, no matter how large, is insufficient for establishing the conjecture's veracity, since a single counterexample would immediately bring down the conjecture.

For example, the Riemann hypothesis is a conjecture from number theory that ( amongst other things ) makes predictions about the distribution of prime numbers.

Because many outstanding problems in number theory, such as Goldbach's conjecture are equivalent to solving the halting problem for special programs ( which would basically search for counter-examples and halt if one is found ), knowing enough bits of Chaitin's constant would also imply knowing the answer to these problems.

Hilbert did not work in the central areas of analytic number theory, but his name has become known for the Hilbert – Pólya conjecture, for reasons that are anecdotal.

The most celebrated single question in the field, the conjecture known as Fermat's Last Theorem, was solved by Andrew Wiles but using tools from algebraic geometry developed during the last century rather than within number theory where the conjecture was originally formulated.

If the four-color conjecture were false, there would be at least one map with the smallest possible number of regions that requires five colors.

In 1973 the number theorist Hugh Montgomery was visiting the Institute for Advanced Study and had just made his pair correlation conjecture concerning the distribution of the zeros of the Riemann zeta function.

For instance, Goldbach's conjecture is the assertion that every even number ( greater than 2 ) is the sum of two prime numbers.

In the second edition of his book on number theory ( 1808 ) he then made a more precise conjecture, with A = 1 and B = − 1. 08366.

For example, the Mertens conjecture is a statement about natural numbers that is now known to be false, but no explicit counterexample ( i. e., a natural number n for which the Mertens function M ( n ) equals or exceeds the square root of n ) is known: all numbers less than 10 < sup > 14 </ sup > have the Mertens property, and the smallest number which does not have this property is only known to be less than the exponential of 1. 59 × 10 < sup > 40 </ sup >, which is approximately 10 to the power 4. 3 × 10 < sup > 39 </ sup >.

The Arnold conjecture on the number of fixed points of Hamiltonian symplectomorphisms and Lagrangian intersections were also a major motivation in the development of Floer homology.

The conjecture is that no matter what number you start with, you will always eventually reach 1.

The Collatz conjecture is: This process will eventually reach the number 1, regardless of which positive integer is chosen initially.

If the conjecture is false, it can only be because there is some starting number which gives rise to a sequence which does not contain 1.

In 1849 de Polignac made the more general conjecture that for every natural number k, there are infinitely many prime pairs p and p ′ such that p ′ − p

A stronger form of the twin prime conjecture, the Hardy – Littlewood conjecture, postulates a distribution law for twin primes akin to the prime number theorem.

This conjecture can be justified ( but not proven ) by assuming that 1 / ln t describes the density function of the prime distribution, an assumption suggested by the prime number theorem.

Polignac's conjecture from 1849 states that for every positive even natural number k, there are infinitely many consecutive prime pairs p and p ′ such that p ′ − p = k ( i. e. there are infinitely many prime gaps of size k ).

Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and in all of mathematics.

Statistical considerations which focus on the probabilistic distribution of prime numbers present informal evidence in favour of the conjecture ( in both the weak and strong forms ) for sufficiently large integers: the greater the integer, the more ways there are available for that number to be represented as the sum of two or three other numbers, and the more " likely " it becomes that at least one of these representations consists entirely of primes.