I compare the efficiency of

x**2 + 2 with the efficiency of

4 x**6 + 4 x**4 - x**2 - 1 = (2 x**2 -1 ) * (2 x**2 + 1) * (x**2 + 1)

for number of iterations needed under Brent's method of factoring.

Almost always the degree 6 polynomial found the factors first.

>>> FactorSix(p7[0])

Factored z = 617999 in 8 steps by Brent Method Two Algorithm.

[409L, 1511L]

>>> FactorSix(p20[0])

Factored z = 62940134090320320101 in 18354 steps by Brent Method Six

Algorithm.

[9942857383L, 6330185747L]

>>> FactorSix(p20[1])

Factored z = 20769381218777181113 in 14216 steps by Brent Method Six

Algorithm.

[2173609013L, 9555251701L]

>>> FactorSix(p20[2])

Factored z = 18692365700052053077 in 36541 steps by Brent Method Six

Algorithm.

[2659137737L, 7029483821L]

>>> FactorSix(p20[3])

Factored z = 24020594249838967723 in 35579 steps by Brent Method Six

Algorithm.

[3027670697L, 7933687859L]

>>> FactorSix(p20[4])

Factored z = 47537628000771577759 in 41656 steps by Brent Method Six

Algorithm.

[5134093499L, 9259205741L]