MathDB

For each

m\in \mathbb N

we define

rad\ (m)=\prod p_i

, where

m=\prod p_i^{\alpha_i}

.
abc Conjecture Suppose

\epsilon >0

is an arbitrary number, then there exist

K

depinding on

\epsilon

that for each 3 numbers

a,b,c\in\mathbb Z

that

gcd (a,b)=1

and

a+b=c

then:

max\{|a|,|b|,|c|\}\leq K(rad\ (abc))^{1+\epsilon}

Now prove each of the following statements by using the

abc

conjecture : a) Fermat's last theorem for

n>N

where

N

is some natural number. b) We call

n=\prod p_i^{\alpha_i}

strong if and only

\alpha_i\geq 2

. c) Prove that there are finitely many

n

such that

n,\ n+1,\ n+2

are strong. d) Prove that there are finitely many rational numbers

\frac pq

such that:

\Big| \sqrt[3]{2}-\frac pq \Big|<\frac{2^ {1384}}{q^3}

Problem Statement