number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

the ''n'' conjecture is a conjecture stated by as a generalization of the ''abc'' conjecture to more than three integers.

Formulations

Given

, let

satisfy three conditions: : (i)

\gcd(a_1,a_2,...,a_n)=1

: (ii)

: (iii) no proper subsum of

equals

First formulation The ''n'' conjecture states that for every

, there is a constant

C

, depending on

and

, such that:

$\operatorname(, a_1, ,, a_2, ,...,, a_n, )< C_\operatorname(, a_1, \cdot , a_2, \cdot ... \cdot , a_n, )^$

where

\operatorname(m)

denotes the

radical Radical may refer to: Politics and ideology Politics *Radical politics, the political intent of fundamental societal change *Radicalism (historical), the Radical Movement that began in late 18th century Britain and spread to continental Europe and ...

of the integer

, defined as the product of the distinct

prime factors A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

. Second formulation Define the ''quality'' of

as :

q(a_1,a_2,...,a_n)= \frac

The ''n'' conjecture states that

\limsup q(a_1,a_2,...,a_n)= 2n-5

Stronger form

proposed a stronger variant of the ''n'' conjecture, where setwise coprimeness of

is replaced by pairwise coprimeness of

. There are two different formulations of this ''strong n'' conjecture. Given

, let

satisfy three conditions: : (i)

are pairwise coprime : (ii)

: (iii) no proper subsum of

equals

First formulation The ''strong n'' conjecture states that for every

, there is a constant

C

, depending on

and

, such that:

$\operatorname(, a_1, ,, a_2, ,...,, a_n, )< C_\operatorname(, a_1, \cdot , a_2, \cdot ... \cdot , a_n, )^$

Second formulation Define the ''quality'' of

as :

q(a_1,a_2,...,a_n)= \frac

The ''strong n'' conjecture states that

\limsup q(a_1,a_2,...,a_n)= 1

References

* * {{DISPLAYTITLE:''n'' conjecture Conjectures Unsolved problems in number theory Abc conjecture