Radical Structure in the MRB Rationality Search

Reduction of the radical n^(1/n)

Suppose the prime factorization of n has prime factors q_1,...,q_r with exponents e_1,...,e_r.

n

r

∏

j=1

e

j

q

j

ggcd(n,

e

1

,…,

e

r

)

d

n

g

c

r

∏

j=1

e

j

g

q

j

The number g is the largest integer that divides n and every prime-factor exponent e_j.

Therefore each exponent e_j/g is an integer, so c is itself a positive integer.

Since every exponent in n is divisible by g, we have:

n

g

c

Also, because d = n/g, we have n = g d. Hence:

1/n

n



1/n

(

g

c

)

1/n

(

g

c

)



g/n

c

g/n

c



1/d

c

1/n

n



1/d

c