Cycles of an Algorithm for the Synthesis of Irreducible Polynomials over the Field $GF(2)$

A study is made of the problem of cycles in the algorithm for the synthesis of irreducible polynomials over the field $GF(2)$ proposed by R. R. Varshamov. A test is also proposed to check for a primitive root with respect to the modulus of a Mersenne prime $n=2^m-1$, requiring less calculation than the usual test.