10
votes
Accepted
Jacobi symbols for two-square sums of primes
The first observation follows from the law of quadratic reciprocity. Indeed, assume that $p\equiv 1\pmod{8}$ and $p=A^2+B^2$. Let $A'$ denote the odd part of $A$. Then $p\equiv B^2\pmod{A'}$, and ...
8
votes
Accepted
Prime differences and zero multiplicity
This problem is connected with the L^2 average of primes in short intervals, see Selberg (1942 paper entitled “on the normal density…”). In particular, results on the integral of $\psi(x+h)-\psi(x)-h$ ...
6
votes
Prime differences and zero multiplicity
It is not know that RH implies EH, or that EH implies RH. Let us denote
$$S(x):=\sum_{p_n < x} (p_n -p_{n-1})^2.$$
Assuming the Lindelöf hypothesis, Yu (1996) proved that $S(x)\ll_\varepsilon x^{1+\...
6
votes
Representing natural numbers as sums of distinct prime powers
Brüdern (2021) proved that every sufficiently large even integer $n$ can be written as
$$n=\sum_{k=1}^{20}x_k^{k+1},$$
where each $x_k$ is a prime number. A similar result with a few more summands was ...
5
votes
Representing natural numbers as sums of distinct prime powers
There's a lot of literature on the topic; the magic words are: "Waring-Goldbach" problem.
Starting point: https://en.wikipedia.org/wiki/Waring%E2%80%93Goldbach_problem
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