The Firoozbakht's conjecture (1982) is equal to:

(p_(n+1))^(n) < (p_n)^(n+1).

Then the natual log is:

n*ln(p_(n+1)) < (n+1)*ln(p_n).

Now,

ln(p_n) <= ln(n) + ln(ln(n)) + 1, for n >= 2. (Dusart 2010)

And, because p_n >= n*ln(n), for n >= 2; (Dusart 1999)

the nat log of p_(n+1) is:

ln(n+1) + ln(ln(n+1)) <= ln(p_(n+1)), for n >= 2.

So,

n*(ln(n+1) + ln(ln(n+1))) < (n+1)*(ln(n) + ln(ln(n)) + 1).

Divided by n*(ln(n) + ln(ln(n)) + 1):

(ln(n+1) + ln(ln(n+1)))/(ln(n) + ln(ln(n)) + 1) < (n+1)/n

This inequality is true because the left-side increases slower than the right-side.

Is this a QED?

John W. Nicholson