Abc conjecture

The abc- conjecture is a 1985 drawn up by Joseph Oesterlé and David Masser mathematical conjecture. It is about the common content of prime factors of triples each other relatively prime natural numbers, in which the third is the sum of the other two is. It describes in a precise form of the phenomenon that the product of all such triples occurring in distinct prime factors is generally not or only slightly smaller than the largest number of the triplet. The additive context of a triplet therefore imposes a strong constraint on the multiplicative structure of the triple figures.

Intuitively based the abc conjecture that natural numbers with numerically many prime factors occur more than once - so-called highly potent or " large numbers " - a relatively rare occurrence. Based on a definition by Barry Mazur, a natural number are referred to as multiplicative highly potent when its binary representation is much longer than the binary representation of its largest square-free divisor, ie the product of all distinct prime factors contained. Then the abc- conjecture states for two relatively prime numbers highly potent and that neither the sum nor their difference can be highly potent, possibly with exceptions, when is small.

The assumption is far neither proved nor disproved, but it applies because of their difficulty and even more so because of its importance as a prominent successor of the dissolved Fermat's theorem (the new " Holy Grail "). Dorian Goldfeld called them even as the most important unsolved problem of Diophantine Analysis. There is already a large number of far-reaching number-theoretic statements known that would follow from the validity of the abc conjecture.

In August 2012, Shinichi Mochizuki published a possible evidence that is currently being examined. It will probably take some time until the confirmation or refutation of the proof, because Mochizuki extensive, recently re- developed by him and so far only applied a few familiar concepts and methods has.

Formulation

A triple is called abc- triple if and are relatively prime positive integers and their sum is. Due to the basic properties divisibility is both to and to prime.

The radical of a positive integer is the product of different prime factors of. Prime factors that occur in the prime factorization of multiple, are included in the calculation of only once. For example, is

Applies to an abc triple the inequality

So it is called abc- hits. Examples are (1, 8, 9 ), (5, 27, 32 ), (32, 49, 81) and the triple of Eric Reyssat found with for which the log ratio ( c) / log (rad (abc) ) = 1.62991 ... is particularly large. abc- hits are rare. Among the 15.2 million abc triples with, there are only 120 abc- hits and among the 380 million abc triples with, there are 276 Sander Dahmen 2006 showed a lower estimate for the number of abc- hits to a given barrier and confirmed the fact that infinitely many exist, but says his formula just under one million abc hit below advance.

The global project ABC @ Home could find so far about 33.18 million abc hit by distributed computing and has set itself the goal of creating a complete list of abc- hits. The list was completed in November 2011. The project was made possible by programming an algorithm that reduced the effort to identify all abc hit with the obvious proportional to almost proportional to computational steps.

Thus, although is usually greater than 1, Masser proved that the ratio can be arbitrarily small. For exponents (even if they come arbitrarily close to 1 ), he formulated but with Oesterlé the abc conjecture that has a positive lower bound.

More precisely is the abc- conjecture:

The conjecture is formulated for, as they like to mention is demonstrably wrong.

One can formulate the conjecture for any positive or negative whole numbers, and then has to replace only on the left side of the inequality by.

Another equivalent formulation of the conjecture is given below.

Implications and variants of the abc- conjecture

Inferences from the abc conjecture

The assumption has not yet though be proved, however, draws a lot of interesting consequences. Many solved and unsolved Diophantine problems can be deduced from this assumption. In particular, the very complex and complicated proof of the Great Fermat's theorem would be reduced to one page.

  • Set of Thue -Siegel -Roth, as Machiel van Frankenhuysen showed 1999.
  • Great Fermat's theorem
  • Conjecture of Mordell ( by Gerd Faltings proved), as Noam Elkies showed 1991. The conjecture asserts the finiteness of the number of points of an algebraic curve of genus greater than 1 over a number field K. From the abc conjecture even follows a bound for the size ( more precisely, the so-called height) of the points on the curves over K ( depending on the constant occurring in the abc- conjecture ). Thus, the abc conjecture gives an effective version of the Mordellvermutung, in contrast to the evidence known to date.
  • Erdős - Woods conjecture
  • Catalansche presumption
  • The existence of infinitely many non- Wieferich primes. General showed Joseph Silverman 1988, that follows from the abc conjecture that there are infinitely many primes for which is not divisible by.
  • The weak form of Hall 's conjecture, which provides an asymptotic lower bound on the amount of the difference of cubes and squares.
  • The presumption of Lucien Szpiro ( an inequality between leaders and discriminant of elliptic curves over the rational numbers ). This assumption is even equivalent to the abc- conjecture.
  • Pillai conjecture of S. S. Pillai.

As an example, the abc conjecture is applied to the great Fermat's theorem that

No solution in positive integers (which are assumed to be relatively prime ) has for

Substituting in the inequality of the abc conjecture, and uses

The inequality is then:

If in this inequality by, then one has for an upper bound on z:

That is, the Fermat may have only finitely many solutions, and at a certain value of the exponent, which depends only on, that would be given by the abc conjecture, no more solution since. One need only all cases up to this limit with other methods to check to prove the Fermatvermutung ( for a large number of exponents was applying the presumption before the proof by Andrew Wiles known).

Special forms of the abc conjecture and weak abc- conjecture

1996 suggested Alan Baker proposes tighter conjecture and clarified it in 2004. During the total size of the multiplicative building blocks of the numbers involved in the triple indicates the number of its distinct prime factors is a measure of their level of detail. Baker united both dimensions and came to abc- conjecture with an absolute, independent, constants

Taking into account the fact that the right side has a minimum at about, and after the replacement in the denominator down estimates by, you get a free version of

Andrew Granville noticed that the last factor is nearly equivalent to Θ (r ), the number of natural numbers to r that are only divisible by prime factors of r. Hence, his presumption may be derived to

A study of the then 196 known extremal abc triples showed that probably and can be selected. Possibly the second value based on recent numerical results must be slightly modified.

There are also weaker forms of the abc- conjecture, one is trying to prove. Is set in the original formulation of the abc conjecture and equal to 1, one has a variant of the weak abc- conjecture ( with the same conditions on the abc- triples as above):

For this variant follows immediately ( by a similar argument as above) the validity of Fermat 's conjecture for powers greater than five. More generally, the weak abc- conjecture is often introduced via a slightly different formulation of the abc- conjecture.

Is the quality (including power, abc- ratio) of (a, b, c ) - triples, that is, the solution of and thus a measure of the growth of c to the common " prime content " r of the cube corner. Extensive numerical search, for example, in ABC @ Home project has been a maximum value of about 1.63 for q result (found by Eric Reyssat so ). A total of only 236 abc- triples could be detected with a quality > 1.4 in 28 years. The actual abc- conjecture, also called strong abc- conjecture then states that

For an arbitrary finite number of solutions has only.

A value of 1 is the best possible lower bound for d Substituting d = 1, there are infinitely many solutions. But even an arbitrarily small value over 1 means after the strong abc- conjecture that the number of solutions is finite.

The weak abc- conjecture states that q has an upper bound. In the special case given above, the upper bound 2 was assumed. From the strong abc- conjecture the validity of the weak abc- conjecture, but does not follow the other way around.

In a symmetrical shape makes the assumption as a statement of the ratio of the height H (a, b, c) = max ( | a |, | b |, | c | ), which measures the size of the numbers involved, to the radical wheel ( a, b, c ) express that measures the prime content. Then says the strong abc- conjecture, that has only finitely many for each prime solutions a, b, c with:

Jeffrey Lagarias and Kannan Soundararajan presented the abc conjecture a "xyz - conjecture " page for the event that all prime factors of the radical of a triplet by a small constant S ( smoothness, smoothness ) are limited, that is. It says that for only finitely many triples abc- exist with. B. de Weger determined this in the results of the ABC @ home project, that of triplet with S = 43 and ( presumably) the largest z as

More reviews of abc- hit

As early as 1986 showed Cameron L. Stewart and Robert Tijdeman that the " quality" assessment of the abc- hits ( with the designations and, )

For growing too fast can not converge to 1, once again, that there is no for. They proved the existence of infinitely many abc triples with

In 2000, stricter M. v. Frankenhuysen this statement with the sets to investigate near whether a given triplet with the evaluation

Exceeds the barrier or not, and to analyze the distribution of the found extremal examples. The following theoretical ( heuristic ) considerations suggest that this review can be unlimited in size to the amount of abc- hits.

For proven results on the distribution of natural numbers with below a given barrier and from ( reasoned and widely confirmed, but unproven ) assumptions about the randomness of the prime factorization in unstructured sets of natural numbers could v. Frankenhuysen stricter lower estimate with a smaller denominator

Derived. Depending on the approach you can select one or one that could not be clarified. The second variant was also found by CL Stewart and G. Tenenbaum (2007, qv). A simple conversion turns them into the elegant review " merit"

Recognize them as analogous to squared with the desired test size.

The current world record for triples with respect to both reviews and was discovered on October 28, 2011 by Ralf Bonse and is

Of particular interest are those ABC triplets which limit the decrease in quality with increasing the amount of downward. An abc- triple is called " unbeaten " ( literally " second to none " ), if any known abc- triples having a smaller larger quality.

Abc- conjecture for polynomials

Wilson Stothers and Richard Mason proved in 1983 independently following, hitherto unknown set of polynomials:

Be coprime, nonconstant polynomials with. Then

The number of distinct zeros is. This is in a sense the " function body " analogue of the abc- conjecture. His proof is relatively simple ( see, eg, Serge Lang, elements of mathematics, Bd.48 (1993 ), pp. 91f ) and, as in the case of the abc- conjecture follows from eg of Fermat's theorem for polynomials. The translation from the polynomial case in the abc- conjecture for integers takes place in that one uses, the product being the " prime " of being, extends over all roots of, and the degree of its analogue replaces the logarithm (da).

However, this " model " version of the abc conjecture was not the immediate motivation for the conjecture by Masser and Oesterlé. The motive for the presumption arose not from numerical calculations, but rather from deep investigations on elliptic curves in number theory, reflected in part in the related conjecture of Lucien Szpiro (see above).

Partial results

So far the following inequalities for c and rad were (a, b, c ) is proved:

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

1991, C. L. Stewart and Kunrui Yu:

1996, C. L. Stewart and Kunrui Yu:

Where C1 is a fixed constant and C2 and C3 positive constants easily computable function of ε.

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