# Number Theory 3 Solutions of 3 Open Problems

Site owner: Li Ke Xiong
Email: likexiong@hotmail.com

The Extreme Solution to Catalan's Conjecture

ABC Solution II      I

A New Interesting Way in Mathematics ---"Relative congruence"

From the basic facts:

1. integer a,b,s. prime the solution always exists and unique (except ). We can use the fractions to express congruence value, directly ,
that is to say the fractions have congruence meaning.

2. So we can make the definition:
(1)
(2) Never can p be denomination divisor in any finite expression, otherwise meaningless.

3. Thus the basic operation can be used on the both sides of without any doubt, it is easy to be proved.

4. Furthermore, we extend the above concept to real number, based on and controlled by power expansion series, that is to say we have discovered the concept of Relative Congruence.

5. Thus we get the necessary condition mod p
for , then we proved mod p never can be satisfied with the same p. .

6. And we have discussed Euler constant c, the conclusion is that c is not rational.   BACK

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