|
The ''abc'' conjecture (also known as the Oesterlé–Masser conjecture) is a conjecture in number theory, first proposed by and . It is stated in terms of three positive integers, ''a'', ''b'' and ''c'' (hence the name), which have no common factors greater than 1 and satisfy ''a'' + ''b'' = ''c''. If ''d'' denotes the product of the distinct prime factors of ''abc'', the conjecture essentially states that ''d'' is usually not much smaller than ''c''. In other words: if ''a'' and ''b'' are composed from large powers of primes, then ''c'' is usually not divisible by large powers of primes. The precise statement is given below. 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". Several solutions have been proposed to the ''abc'' conjecture, the most recent of which is still being evaluated by the mathematical community. ==Formulations== The abc conjecture can be expressed as follows: For every ε > 0, there are only finitely many triples of coprime positive integers ''a'' + ''b'' ''c'' such that ''c'' > ''d''1+ε, where ''d'' denotes the product of the distinct prime factors of ''abc''. To illustrate the terms used, if :''a'' = 16 = 24, :''b'' = 17, and :''c'' = 16 + 17 = 33 = 3·11, then ''d'' = 2·17·3·11 = 1122, which is greater than ''c''. Therefore, for all ε > 0, ''c'' is not greater than ''d''1+ε. According to the conjecture, most coprime triples where are like the ones used in this example, and for only a few exceptions is ''c'' > ''d''1+ε. To add more terminology: For a positive integer ''n'', the radical of ''n'', denoted rad(''n''), is the product of the distinct prime factors of ''n''. For example * rad(16) = rad(24) = 2, * rad(17) = 17, * rad(18) = rad(2·32) = 2·3 = 6. If ''a'', ''b'', and ''c'' are coprime〔When ''a'' + ''b'' = ''c'', coprimeness of ''a'', ''b'', ''c'' implies pairwise coprimeness of ''a'', ''b'', ''c''. So in this case, it does not matter which concept we use.〕 positive integers such that ''a'' + ''b'' = ''c'', it turns out that "usually" ''c'' < rad(''abc''). The ''abc conjecture'' deals with the exceptions. Specifically, it states that: ABC Conjecture. For every ε > 0, there exist only finitely many triples (''a'', ''b'', ''c'') of positive coprime integers, with ''a'' + ''b'' = ''c'', such that An equivalent formulation states that: ABC Conjecture II. For every ε > 0, there exists a constant ''K''ε such that for all triples (''a'', ''b'', ''c'') of coprime positive integers, with ''a'' + ''b'' = ''c'', the inequality A third equivalent formulation of the conjecture involves the ''quality'' ''q''(''a'', ''b'', ''c'') of the triple (''a'', ''b'', ''c''), defined as : For example, * ''q''(4, 127, 131) = log(131) / log(rad(4·127·131)) = log(131) / log(2·127·131) = 0.46820... * ''q''(3, 125, 128) = log(128) / log(rad(3·125·128)) = log(128) / log(30) = 1.426565... A typical triple (''a'', ''b'', ''c'') of coprime positive integers with ''a'' + ''b'' = ''c'' will have ''c'' < rad(''abc''), i.e. ''q''(''a'', ''b'', ''c'') < 1. Triples with ''q'' > 1 such as in the second example are rather special, they consist of numbers divisible by high powers of small prime numbers. ABC Conjecture III. For every ''ε'' > 0, there exist only finitely many triples (''a'', ''b'', ''c'') of coprime positive integers with ''a'' + ''b'' = ''c'' such that ''q''(''a'', ''b'', ''c'') > 1 + ''ε''. Whereas it is known that there are infinitely many triples (''a'', ''b'', ''c'') of coprime positive integers with ''a'' + ''b'' = ''c'' such that ''q''(''a'', ''b'', ''c'') > 1, the conjecture predicts that only finitely many of those have ''q'' > 1.01 or ''q'' > 1.001 or even ''q'' > 1.0001, etc. In particular, if the conjecture is true then there must exist a triple (''a'', ''b'', ''c'') which achieves the maximal possible quality ''q''(''a'', ''b'', ''c'') . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「abc conjecture」の詳細全文を読む スポンサード リンク
|