What happens to the Pollard rho algorithm when it is run wit

What happens to the Pollard rho algorithm when it is run with n = 35, x_1 = 6? Show that the same thing happens (for at least one value of x_1) whenever n = p(p + 2) for twin primes p,p + 2.

Solution

Polaards rho algorithm is a special purpose integer factorization algorithm.It is effective composite number. The algorithm takes the inputs as n the integer to be factored. g(x) a polynomial p(x) computed modulo n. if p/n and x is congruent to y modp then g(x) is congruent to g(y) modp. It holds for g(x)=p modn and also for g(x)=p+2 mod n.We have that rho algorithmcomes from the fact that the values of p modn eventually repeat the period resulting in a rho shape in a graph of the values.

let p and q=p+2 be any two consecutive twin primes then the product pq can be given by (n-1)(n+1). consider a composite integer m=pq that is a product of twin primes then.

(n-1)(n+1)=m

n² -1=m and n=(m+1)^1/2

^{taking the appropriate value of n to be approximately p and q}