n conjecture

Template:Short description Template:Technical

In number theory, the n conjecture is a conjecture stated by Template:Harvtxt as a generalization of the abc conjecture to more than three integers.

Formulations

Given $n \geq 3$ , let $a_{1}, a_{2}, . . ., a_{n} \in ℤ$ satisfy three conditions:

(i)

\gcd (a_{1}, a_{2}, . . ., a_{n}) = 1

(ii)

a_{1} + a_{2} + . . . + a_{n} = 0

(iii) no proper subsum of

a_{1}, a_{2}, . . ., a_{n}

equals

0

First formulation

The n conjecture states that for every $ε > 0$ , there is a constant $C$ depending on $n$ and $ε$ , such that:

$\max (| a_{1} |, | a_{2} |, . . ., | a_{n} |) < C_{n, ε} rad (| a_{1} | \cdot | a_{2} | \cdot \dots \cdot | a_{n} |)^{2 n - 5 + ε}$

where $rad (m)$ denotes the radical of an integer $m$ , defined as the product of the distinct prime factors of $m$ .

Second formulation

Define the quality of $a_{1}, a_{2}, . . ., a_{n}$ as

q (a_{1}, a_{2}, . . ., a_{n}) = \frac{\log (\max (| a_{1} |, | a_{2} |, . . ., | a_{n} |))}{\log (rad (| a_{1} | \cdot | a_{2} | \cdot . . . \cdot | a_{n} |))}

The n conjecture states that $lim sup q (a_{1}, a_{2}, . . ., a_{n}) = 2 n - 5$ .

Stronger form

Template:Harvtxt proposed a stronger variant of the n conjecture, where setwise coprimeness of $a_{1}, a_{2}, . . ., a_{n}$ is replaced by pairwise coprimeness of $a_{1}, a_{2}, . . ., a_{n}$ .

There are two different formulations of this strong n conjecture.

Given $n \geq 3$ , let $a_{1}, a_{2}, . . ., a_{n} \in ℤ$ satisfy three conditions:

(i)

a_{1}, a_{2}, . . ., a_{n}

are pairwise coprime

(ii)

a_{1} + a_{2} + . . . + a_{n} = 0

(iii) no proper subsum of

a_{1}, a_{2}, . . ., a_{n}

equals

0

First formulation

The strong n conjecture states that for every $ε > 0$ , there is a constant $C$ depending on $n$ and $ε$ , such that:

$\max (| a_{1} |, | a_{2} |, . . ., | a_{n} |) < C_{n, ε} rad (| a_{1} | \cdot | a_{2} | \cdot \dots \cdot | a_{n} |)^{1 + ε}$

Second formulation

Define the quality of $a_{1}, a_{2}, . . ., a_{n}$ as

q (a_{1}, a_{2}, . . ., a_{n}) = \frac{\log (\max (| a_{1} |, | a_{2} |, . . ., | a_{n} |))}{\log (rad (| a_{1} | \cdot | a_{2} | \cdot . . . \cdot | a_{n} |))}

The strong n conjecture states that $lim sup q (a_{1}, a_{2}, . . ., a_{n}) = 1$ .

References

n conjecture

Formulations

Stronger form

References

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