I checked some relations between primes, here $1<n<10^5$ and $p\_n$ is the $n$th prime.


$a) p\_n^{1/3} - p\_{n-1}^{1/3}<1/2$


$b) p\_n^{1/n} - p\_{n-1}^{1/n}<1/n $


$c) (\log p\_n)^{1/2} - (\log p\_{n-1})^{1/2} < 1/4$


$d) (\log p\_n)^{1/n} - (\log p\_{n-1})^{1/n} < 1/n^X, n\geq7,X=2$


In $d)$ I tried to find a larger $X$, but I failed.


Maybe some will fail for a larger $n$. Are any of these know to be true? Also, what can be deduced from they?



Some conjectures about prime gaps
The first three statements are true for $n$ sufficiently large. The fourth statement is equivalent to a very strong bound on prime gaps (which has a chance to hold but is right on the edge and hopeless to prove or disprove at present), while any $X>2$ produces a false bound for all $n>n\_0(X)$.


Regarding a), a classical result of Ingham (1937) shows that
$$p\_n^{1/3} - p\_{n-1}^{1/3}<\frac{p\_n-p\_{n-1}}{3p\_{n-1}^{2/3}}=o(1).$$


Regarding b), the Prime Number Theorem gives that
$$p\_n^{1/n} - p\_{n-1}^{1/n}\sim\frac{p\_n-p\_{n-1}}{n p\_{n-1}}=o\left(\frac{1}{n}\right).$$


Regarding c), Bertrand's postulate shows for $p\_n>7$ that
$$(\log p\_n)^{1/2}-(\log p\_{n-1})^{1/2}<\frac{\log p\_n-\log p\_{n-1}}{2(\log p\_{n-1})^{1/2}}<\frac{\log 2}{2(\log p\_{n-1})^{1/2}}<\frac{1}{4}.$$
In fact the left-hand side tends to zero (as $n\to\infty$).


Regarding d), the Prime Number Theorem gives that
$$(\log p\_n)^{1/n}-(\log p\_{n-1})^{1/n}\sim\frac{p\_n-p\_{n-1}}{n p\_n\log p\_n}\sim\frac{p\_n-p\_{n-1}}{n^2\log^2 n}.$$
Under some standard (but bold) conjectures the right-hand side stays below $2/n^2$, and perhaps it is even $o(1/n^2)$. So statement d) restricted to large $n$ is equivalent to some expected but hopelessly difficult upper bound on prime gaps. On the other hand, the above calculation shows that d) fails for any $X>2$ and $n>n\_0(X)$.


