Problems & Puzzles: Conjectures

Conjecture 76. Conjectures related to sequence (P[n])^(1/n)

Mahdi Meisami sent the following conjectures:

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))

Q. Prove them or find counterexamples.

Alexei Kourbatov wrote on March 9, 2016

Regarding Conjecture 76: All of these statements follow from Firoozbakht's conjecture 30.

Firoozbakht's conjecture is true for primes up to 4*10^18; see http://arxiv.org/abs/1503.01744.

Therefore:
- statements involving n*m are true at least for n*m<4*10^18  (Conj1, Conj2, Stronger-Conj1).

- statements involving n+m are true at least for n+m<4*10^18  (Conj3, Conj4, Stronger-Conj2).

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