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I am learning Pollard rho algorithm, and I cannot understand end loop condition. As I understand, this algorithm can be applied for number $N$, if we know, that such number has at least one prime factor $p : 1 < p < N$? If it is so, what is the benefit of such algorithm? It turns out that we always should check, whether number has at least one not trivial prime factor.

P.S. As I understand, end loop condition of algorithm is flag, indicating whether we have found prime factor. In other words, without "composite number" check, inifite loops are possible?

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You never apply Pollard $\rho$ to a number unless you have already proved that the number is composite. But there are ways to prove a number is composite that are far, far faster than looking for a prime factor of the number. Look up "Fermat's Little Theorem".

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    I thought about such approach, but I suppose that it is not trivial task. What about testing big number? I doubt that such approach will give good results in reasonbly short time – 2017-02-26
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    See the highest-voted answer to http://stackoverflow.com/questions/2267146/what-is-the-fastest-factorization-algorithm – 2017-02-26