Are all values of the totient function for composite numbers between two consecutive primes smaller than the totient values of those primes? I've seen the graph for $\varphi(n)$ vs $n$ on Wolfram, and it seems like the $\varphi(n)$ values for primes follow a constant slope, but is there a proof that states that the totient values between two consecutive primes are always smaller than the totient values of those two primes? EDIT: I got a satisfactory answer for the case where there is a strict inequality. But what is the condition for a relaxed inequality? Is there a proof that establishes $\varphi(n)\le\varphi(p_n)$ and $\varphi(n)\le\varphi(p_{n+1})$

Let's assume that Oppermann's conjecture is true and for every prime $p_n$ and next prime $p_{n+1}$,$$p_{n+1}
Now note that every number $k$ such that $p_n
EDIT: One can verify or find counterexamples of the inequality manually for small $k$s around above counterexamples.

If Oppermann's conjecture is not true, and furthermore, there exists a prime $q$ and integer $n$ such that $p_np_n$$so your conjecture has counterexample.