## 5 Reputation

16 years, 200 days

## Factorization of N=pq are partially in P...

Maple 2020

I found a condition for p, q that N=pq can be factored in plynominal time using Maple 2020.
Is fllowing Hypothesis and Proof is right?

Hypothesis

N=pq  p and q are large prime respectively.
R=q/p  q > p  R is very close to an small integer or a simple rational number.

N=pq can be factorized in time polynomial

Proof
point[p, q] is on y=N/x
y=N/x  and y=Rx cross at point[p, q]
N is n digit
upper  2 digits N2  round off the 3rd digit
upper  3 digits N3  round off the 4th digit
upper  4 digits N4  round off the 5th digit

y=N2/x and y=Rx cross at point[p2,q2]
y=N3/x and y=Rx cross at point[p3,q3]
y=N4/x and y=Rx cross at point[p4, p4]

But we only know N.

Let line up candidates point[p2,q2] , point[p3,q3] and point[p4, p4]

N2 < 99  i=1..10 j=1..10
R2=i/j
f2=N2/R2 - j^2
dn2=abs(N2-R2*j^2)

N4 < 9999  i=1..99 j=1.. sqrt(N4)
R4=i/j
f4=N4/R4 - j^2
dn4=abs(N4-R4*j^2)

Point[j, i] that have  small f2 and dn2 can be nominated as candidate for point[p2, q2]
Point[j, i] that have  small f3 and dn3 can be nominated as candidate for point[p3, q3]
Point[j, i] that have  small f4 and dn4 can be nominated as candidate for point[p4, q4]

Find cross point[px, qx] of y=R2x and y=N/x , y=R3x, y=N/x and y=R4x, y=N/x
Find the nearest prime pn for px and the nearest prime qn for qx

pn*qn=N  bingo!

Number of candidates are finit.
You can factorized N=pq in time polynomial.

Q.E.D. ?

Rang from p - half digits of p to p + half degits of p and /or range q - half digits of q to q + half degits of q  N=pq can be factored in plynominal time.

## Strange Factorization...

Maple

At first, I appology my poor English and I am a non-proessional on mathematics and programing. I am an old retired engineer/researcher.

I found strange phenomena with maple.
Could someone please verify this result?
This is sound like a stupid joke.
This queation is related to factorization of composite number N=p*q
p, q are large prime number respectively.

Degits:=200;
1.    Let p is large prime such as 50 degits
2.    Let R is small integer such as 3
3.    Let q is nearest prime R*p
4.    Let N=p*q
5.    Let f=frac(N/(q+s))

j:=1;
m:=2^b*2^c;
r:=m/2^b;
b:=8; c:=0;
F:=[ ];
for s from -m to m by r do
f:= frac(N/(q+s));
if f=0 OR 1.0 then f:=100000; # zero dev exeption;
elif f <= 0.5 then f:=log10(f);
elif f > 0.5 AND f < 1.0 then f:=-log10(1.0-f);
else print(“error”) break;
end if:
F:=[F[ ], [s, f]];  # maple plot format
j::=j+1;
end do:

6.    Draw a Graph s vs F

You wil find a oscillating graph with step at s=0;

7. b:=8 and c:=75; and try same caluclation.

You will find a graph with dip at s=0;
Caution! Resolution and Scanwidth can not go together.

8.    Let qx=q+10^25;
9.    Nx:=p*qx;
10.    Let ps:=sqrt(N/3.0) and let qs:=3.0*sqrt(N/3.0);
11.    fs:=frac(Nx/(qx+s));
Caution!  Nx not equal Ns:=p*qs
12.     Draw a graph s vs Fs with Nx, p, qs
You will find a similar graph dip point shifted from s=0
13. Verify qx=ps-dip point s

My questions are
1.    Is it a kaind of factorization of Nx?
2.    This phenomena hava posibirity to make a vulneravility for RSA crypto?
3.    Power spectrum of F( c=0) indicates R=q/p; with an peak
4.    Is it an rediscovery of “Wheel”?  Are thea any papers similar to this phenomena

I appreciate to read this questions.

Environment: Windows 10 for workstation and Maple 2019(64bit) CPU: Intel Xeon.

15 Dec 2019  some miss spelling corrected

A graph added   frac(N(q+s)) vs s b=8 c=75

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