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Old 2021-07-28, 20:21   #9
RomanM
 
Jun 2021

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Ok!
1. Code find the values of t>sqrt(p) (p - any number. Can be prime or composite), for those mod(t^2,p)<sqrt(p) and do this in a very unusual way, far away from common approach.
Algorithm is quite simple.

Take some integer u>sqrt(p), b=mod(u^2,p); a=mod(b^2,p)=mod(u^4,p);

[From (b-y)^2==0 mod p
b^2-2*b*y+y^2==0 mod p or a-2*b*y+y^2==0;
Solution: y=b-sqrt(b^2-a) (and y=b+sqrt(b^2-a), using first)
Make y an integer, and compute t= b-y =ceil(sqrt(b^2-a))]

So t=ceil(sqrt(b^2-a)). t is some integer) Let u=t, and go all this again, in cycle.
After few step, the value of b became less than sqrt(p) (or cycle go to some ring)

2. See 3.
3. why the hell does this even work???
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