On the sum of a prime and a k-th power of prime in short intervals

Wang, Y.
May 2012
Acta Mathematica Hungarica;May2012, Vol. 135 Issue 3, p248
Academic Journal
Let $\mathcal{H}_{k}$ denote the set { n∣2| n, $n\not\equiv 1\ (\mathrm{mod}\ p)$ ∀ p>2 with p−1| k}. We prove that when $X^{\frac{11}{20}\left(1-\frac{1}{2k}\right) +\varepsilon}\leqq H\leqq X$, almost all integers $n\in\allowbreak {\mathcal{H}_{k} \cap (X, X+H]}$ can be represented as the sum of a prime and a k-th power of prime for k≧3. Moreover, when $X^{\frac{11}{20}\left(1-\frac{1}{k}\right) +\varepsilon}\leqq H\leqq X$, almost all integers n∈( X, X+ H] can be represented as the sum of a prime and a k-th power of integer for k≧3.


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