20th Test of primality and factorization of Lepore with Pythagorean triples

[Alberico Lepore]

20th Test of primality and factorization of Lepore with Pythagorean triples
(conjecture) in linear coputational complexity


What do you think about it?

[jnml]

Originally Posted by Alberico Lepore

20th Test of primality and factorization of Lepore with Pythagorean triples
(conjecture) in linear coputational complexity


What do you think about it?

Copy of the document body:

Sian N = p * q with p and q integer then there will be a Pythagorean triplet,
with a smaller cateto N and the other two sides C and D (respectively cateto
and hypotenuse),such that GCD (N, C, D) = p or GCD (N, C, D) = q.

Therefore, having a table with the Pythagorean triples ordered by a minor
cateto will be able to factor or establish primality in linear computational
complexity.

[Alberico Lepore]

additionally

N^2+C^2=D^2 , (C+D)/q=p^2 , D-C=q
and
N^2+C^2=D^2 , (C+D)/p=q^2 , D-C=p

[LaurV]

Originally Posted by jnml

Therefore, having a table with the Pythagorean triples ordered by a minor
cateto will be able to factor or establish primality in linear computational
complexity.

Leaving apart the fact that this ignores the time to search in the table, I have a better version:

"Having a table with Natural Numbers N ordered by N, and their factorization will be able to factor or establish primality in linear computational complexity". Why do you need Pythagorean triples?

[CRGreathouse]

Originally Posted by LaurV

"Having a table with Natural Numbers N ordered by N, and their factorization will be able to factor or establish primality in linear computational complexity". Why do you need Pythagorean triples?

I have a method by which I can construct the n-th natural number directly, obviating the need for initialization and storage. Combining our technologies, we could get nearly the efficiency of trial division with just as little memory.

[Alberico Lepore]

Originally Posted by LaurV

Leaving apart the fact that this ignores the time to search in the table, I have a better version:

"Having a table with Natural Numbers N ordered by N, and their factorization will be able to factor or establish primality in linear computational complexity". Why do you need Pythagorean triples?

Because with the Pythagorean triples I found a method to eliminate a factor.
Only I have to get back to solving this (2077*(4*sqrt(2*b+1)-3))/(32*b+7)=q
can you help me?

[CRGreathouse]

Originally Posted by Alberico Lepore

Because with the Pythagorean triples I found a method to eliminate a factor.
Only I have to get back to solving this (2077*(4*sqrt(2*b+1)-3))/(32*b+7)=q
can you help me?

b = (4313929 - 12462q - 7q^2)/(32q^2).

[Alberico Lepore]

Originally Posted by CRGreathouse

b = (4313929 - 12462q - 7q^2)/(32q^2).

what procedure is used to solve it?

[jnml]

Originally Posted by Alberico Lepore

what procedure is used to solve it?

For example.

[Alberico Lepore]

Originally Posted by jnml

For example.

i do not have PRO

[CRGreathouse]

Originally Posted by Alberico Lepore

i do not have PRO

The free version should do it. Failing that, you could just, you know, solve the equation?