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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 $theta = (log q) / log x$.
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
It is proven that there are infinitely prime pairs whose difference is no greater than 20.
We determine primitive solutions to the equation $(x-r)^2 + x^2 + (x+r)^2 = y^n$ for $1 le r le 5,000$, making use of a factorization argument and the Primitive Divisors Theorem due to Bilu, Hanrot and Voutier.
The idea of generating prime numbers through sequence of sets of co-primes was the starting point of this paper that ends up by proving two conjectures, the existence of infinitely many twin primes and the Goldbach conjecture. The main idea of our ap
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 $loglog xge 15$.