We give an explicit form of Inghams Theorem on primes in the short intervals, and show that there is at least one prime between every two consecutive cubes $xsp{3}$ and $(x+1)sp{3}$ if $loglog xge 15$.
In this paper, we consider the problem about finding out perfect powers in an alternating sum of consecutive cubes. More precisely, we completely solve the Diophantine equation $(x+1)^3 - (x+2)^3 + cdots - (x + 2d)^3 + (x + 2d + 1)^3 = z^p$, where $p
$ is prime and $x,d,z$ are integers with $1 leq d leq 50$.
We use Maynards methods to show that there are bounded gaps between primes in the sequence ${lfloor nalpharfloor}$, where $alpha$ is an irrational number of finite type. In addition, given a superlinear function $f$ satisfying some properties describ
ed by Leitmann, we show that for all $m$ there are infinitely many bounded intervals containing $m$ primes and at least one integer of the form $lfloor f(q)rfloor$ with $q$ a positive integer.
Let $t in mathbb{N}$, $eta >0$. Suppose that $x$ is a sufficiently large real number and $q$ is a natural number with $q leq x^{5/12-eta}$, $q$ not a multiple of the conductor of the exceptional character $chi^*$ (if it exists). Suppose further that,
[ max {p : p | q } < exp (frac{log x}{C log log x}) ; ; {and} ; ; prod_{p | q} p < x^{delta}, ] where $C$ and $delta$ are suitable positive constants depending on $t$ and $eta$. Let $a in mathbb{Z}$, $(a,q)=1$ and [ mathcal{A} = {n in (x/2, x]: n equiv a pmod{q} } . ] We prove that there are primes $p_1 < p_2 < ... < p_t$ in $mathcal{A}$ with [ p_t - p_1 ll qt exp (frac{40 t}{9-20 theta}) . ] Here $theta = (log q) / log x$.