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++
}
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