range ok h >=1


I tried to write the pseudo code

[CODE]i=0

while(i<10) {

solve this system and memorize y(i) and r(i)

sqrt(N/((10+i)/10))=a
,
((10+i)/10*a+a-4)/8=x
,
2*x*(x+1)-y*(y-1)/2=(N-3)/8
,
(sqrt(32*x+1)+1)/2=b
,
b*(b-1)/2-(sqrt(32*(x-b)+1)+1)/2*[(sqrt(32*(x-b)+1)+1)/2-1]/2=r

}

j=0

while (!(N mod p ==0 && p!=1 && p!=N)){

i=0

while(i<10) {

solve this system with unique integer solution of h

2*(h)*(h-1)<(N-3)/8+k*(k-1)/2<=2*(h)*(h+1)
,
2*(x)*(x+1)-y*(y-1)/2=(N-3)/8
,
x-(sqrt(32*x+1)+1)/2<h<x+(sqrt(32*x+1)+1)/2
,
k=y(i)+j*r(i)

in the range [y(i)+(j-1)*r(i),y(i)+j*r(i)] in log_2 search the point if the system admit solutions

if you find it {

x-(sqrt(32*x+1)+1)/2=h

aproximate x to integer

2*(x)*(x+1)-y*(y-1)/2=(N-3)/8

calculate p

p=4*x+1-2*(y-1)
}
i++
}

j++
}[/CODE]

(*) here I have to improve, multiplying the range by 10 ^ (tot), but I still don't know tot

[CODE]
i=0

while(i<10) {

solve this system and memorize y(i) and r(i)

sqrt(N/((10+i)/10))=a
,
((10+i)/10*a+a-4)/8=x
,
2*x*(x+1)-y*(y-1)/2=(N-3)/8
,
(sqrt(32*x+1)+1)/2=b
,
[b*(b-1)/2-(sqrt(32*(x-b)+1)+1)/2*[(sqrt(32*(x-b)+1)+1)/2-1]/2]/2=r

}

j=0

while (!(N mod p ==0 && p!=1 && p!=N)){

i=0

while(i<10) {

solve this system with unique integer solution of h

2*(h)*(h-1)<(N-3)/8+k*(k-1)/2<=2*(h)*(h+1)
,
2*(x)*(x+1)-y*(y-1)/2=(N-3)/8
,
x-(sqrt(32*x+1)+1)/2<h<x+(sqrt(32*x+1)+1)/2
,
k=y(i)+j*r(i)

in the range [y(i)+(j-2)*r(i),y(i)+j*r(i)] in log_2 search 2>[range of x]>=1 if exist (*)

if exist {

choose the only possible integer solution of x

x-(sqrt(32*x+1)+1)/2=h


2*(x)*(x+1)-y*(y-1)/2=(N-3)/8

calculate p

p=4*x+1-2*(y-1)
}
i++
}

j++
}[/CODE]






Example


N=390644893234047643
,
sqrt(N/(15/10))=a
,
(15/10*a+a-4)/8=x
,
2*x*(x+1)-y*(y-1)/2=(N-3)/8
,
(sqrt(32*x+1)+1)/2=b
,
[b*(b-1)/2-(sqrt(32*(x-b)+1)+1)/2*[(sqrt(32*(x-b)+1)+1)/2-1]/2]/2=r

r=71437,.....




N=390644893234047643
,
2*(h)*(h-1)<(N-3)/8+k*(k-1)/2<=2*(h)*(h+1)
,
2*(x)*(x+1)-y*(y-1)/2=(N-3)/8
,
x-(sqrt(32*x+1)+1)/2<h<x+(sqrt(32*x+1)+1)/2
,
k=63790420+j*71437

suppose we have arrived at j =34




k in the range [63790420+32*71437;63790420+34*71437]

with biinarie research find x=159757905

for k=66207577

N=390644893234047643
,
2*(h)*(h-1)<(N-3)/8+k*(k-1)/2<=2*(h)*(h+1)
,
2*(x)*(x+1)-y*(y-1)/2=(N-3)/8
,
x-(sqrt(32*x+1)+1)/2<h<x+(sqrt(32*x+1)+1)/2
,
k=66207577

infatti range x (159757904,46492;159757905,46503)

range size >= 1 & <2

If we establish how far x can be from the [COLOR="Red"]capofila[/COLOR] at mos (order size)t, here

r=71501
,
r=(2*h+sqrt(32*h+81)+9)/2-(2*h-sqrt(32*h+49)+7)/2
,
(2*x-sqrt(32*x+1)-1)/2<h<(2*x+sqrt(32*x+1)+1)/2


[COLOR="red"]159757809[/COLOR]<x<159793564

in this case 95


we will establish the maximum time



N=390644893234047643 , sqrt(N)=a , (a+a-4)/8=x , (sqrt(32*x+1)+1)/2=b/2

b=70712

(71501-70712)*distance from the [COLOR="red"]capofila[/COLOR]

789*distance from the [COLOR="red"]capofila[/COLOR]

[CODE]
check=0

i=0

while(i<10) {

solve this system

sqrt(N/((10+i)/10))=a
,
((10+i)/10*a+a-4)/8=x
,
2*x*(x+1)-b*(b-1)/2=(N-3)/8

}

memorize b(i)


C=0

while (check==0){

i=0

while(i<10 && check==0) {

h=x-b(i)/2-C // b/2 must be integer
,
[2*(h-1)*(h-1+1)]
<
N-3)/8-b(i)/2*(4*x+1-2*(y-1))
<=
[2*h*(h+1)]
,
2*(x)*(x+1)-y*(y-1)/2=(N-3)/8


while(min_h <= max_h && check==0){

x=h+b/2+C

2*(x)*(x+1)-y*(y-1)/2=(N-3)/8

calculate p

p=4*x+1-2*(y-1)

if(N mod p ==0 && p!=1 && p!=N){
check=1
break

}

min_h++

}
i++
}

C++
}[/CODE]




Example

N=390644893234047643
,
sqrt(N/(15/10))=a
,
(15/10*a+a-4)/8=x
,
2*x*(x+1)-b*(b-1)/2=(N-3)/8

b = 63790420






h=x-(63790420)/2-C
,
[2*(h-1)*(h-1+1)]
<
(390644893234047643-3)/8-(63790420)/2*(4*x+1-2*(y-1))
<=
[2*h*(h+1)]
,
2*(x)*(x+1)-y*(y-1)/2=(390644893234047643-3)/8


per C=-7454

127855236<=h<127855255 -> size range = 19


per h=127855241
,
h=x-(63790420)/2-7454

->

x=159757905



the problem is that size range of h is decreasing

for C=0

127580838<=h<127584034 -> size range = 3196

for C=3727

127775161<=h<127775188 -> size range =27


an exponential decrease would seem to our advantage




*********************************************************************

UPDATE:

I tried to solve in x and I noticed that the first valid value is our x = 159757905
if
it would always happen

the cost of factoring 390644893234047643 would be 7454 * 10 = 74540

Tomorrow morning I will continue with other tests

[url]https://www.wolframalpha.com/input/?i=solve+h%3Dx-%2863790420%29%2F2-7454+%2C+%5B2*%28h-1%29*%28h-1%2B1%29%5D+++%3C++%28390644893234047643-3%29%2F8-%2863790420%29%2F2*%284*x%2B1-2*%28y-1%29%29++%3C%3D+++%5B2*h*%28h%2B1%29%5D++%2C+++2*%28x%29*%28x%2B1%29-y*%28y-1%29%2F2%3D%28390644893234047643-3%29%2F8+%2Ch+%2Cy[/url]

[QUOTE=Alberico Lepore;584549]

*********************************************************************

UPDATE:

I tried to solve in x and I noticed that the first valid value is our x = 159757905
if
it would always happen

the cost of factoring 390644893234047643 would be 7454 * 10 = 74540

Tomorrow morning I will continue with other tests

[url]https://www.wolframalpha.com/input/?i=solve+h%3Dx-%2863790420%29%2F2-7454+%2C+%5B2*%28h-1%29*%28h-1%2B1%29%5D+++%3C++%28390644893234047643-3%29%2F8-%2863790420%29%2F2*%284*x%2B1-2*%28y-1%29%29++%3C%3D+++%5B2*h*%28h%2B1%29%5D++%2C+++2*%28x%29*%28x%2B1%29-y*%28y-1%29%2F2%3D%28390644893234047643-3%29%2F8+%2Ch+%2Cy[/url][/QUOTE]

unfortunately this is not true.


But x is very close to min_range_x

I tested on a number of 30 digits with p and q of 15 digits and the result is 37

N=188723059539473758658629052963


N=188723059539473758658629052963
,
sqrt(N/(18/10))=a
,
(18/10*a+a-4)/8=x
,
2*x*(x+1)-b*(b-1)/2=(N-3)/8

b=64759908643727


h=x-(64759908643726)/2-88973930
,
[2*(h-1)*(h-1+1)]
<
(188723059539473758658629052963-3)/8-(64759908643726)/2*(4*x+1-2*(y-1))
<=
[2*h*(h+1)]
,
2*(x)*(x+1)-y*(y-1)/2=(188723059539473758658629052963-3)/8

range x 113364197263548<=x<=113364197263741

size_range 193

distance x 37

x=113364197263585

I need to be able to quantify distance x or size_range_x

total cost [10*my_quantify *88973930] about N ^ (1/3)

It's still a very heavy bruteforce, but I'm happy