For every natural number n & m we
have :

Conj1: (P[n]^(1/n))*(P[m]^(1/m)) >
P[n*m]^(1/(n*m))

Conj2: (P[n]^(1/n)) + (P[m]^(1/m))
> P[n*m]^(1/(n*m))

Conj3: (P[n]^(1/n))*(P[m]^(1/m)) >
P[n + m]^(1/(n + m))

Conj4: (P[n]^(1/n)) + (P[m]^(1/m))
> P[n + m]^(1/(n + m))

where P[n] is the nth prime
number.

Also we can make conj2 & con4 more
stronger in this way:

Stronger-Conj2: (P[n]^(1/n)) + (P[m]^(1/m))>1+P[n*m]^(1/(n*m))

Stronger-Conj4: (P[n]^(1/n)) + (P[m]^(1/m))>1+P[n
+ m]^(1/(n + m))