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Old 2021-07-30, 17:55   #14
Alberico Lepore
 
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May 2017
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22·127 Posts
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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
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