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

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


What do you think about it?

[QUOTE=Alberico Lepore;476955]20th Test of primality and factorization of Lepore with Pythagorean triples
(conjecture) in linear coputational complexity


What do you think about it?[/QUOTE]

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.

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

[QUOTE=jnml;476956]
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.[/QUOTE]
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?

[QUOTE=LaurV;476977]"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?[/QUOTE]

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. :grin:

[QUOTE=LaurV;476977]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?[/QUOTE]

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?

[QUOTE=Alberico Lepore;477045]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?[/QUOTE]

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

[QUOTE=CRGreathouse;477051]b = (4313929 - 12462q - 7q^2)/(32q^2).[/QUOTE]

what procedure is used to solve it?

[QUOTE=Alberico Lepore;477059]what procedure is used to solve it?[/QUOTE]

[URL="https://www.wolframalpha.com/input/?i=solve+(2077*(4*sqrt(2*b%2B1)-3))%2F(32*b%2B7)%3Dq,+b"]For example[/URL].

[QUOTE=jnml;477061][URL="https://www.wolframalpha.com/input/?i=solve+(2077*(4*sqrt(2*b%2B1)-3))%2F(32*b%2B7)%3Dq,+b"]For example[/URL].[/QUOTE]

i do not have PRO

[QUOTE=Alberico Lepore;477063]i do not have PRO[/QUOTE]

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