Twin Primes and a Primality Test by Indivisibility

Multidimensional Schinzel-type theorems for multiplicative functions near prime tuples

Assuming Dickson's conjecture, we obtain multidimensional analogues of recent results on the behavior of certain multiplicative arithmetic functions near twin-prime arguments. This is inspired by analogous unconditional theorems of Schinzel undertaken without primality assumptions.

Quantum Computation of Prime Number Functions

We propose a quantum circuit that creates a pure state corresponding to the quantum superposition of all prime numbers less than 2^n, where n is the number of qubits of the register. This Prime state can be built using Grover's algorithm, whose oracle is a quantum implementation of the classical Miller-Rabin primality test. The Prime state is highly entangled, and its entanglement measures encode number theoretical functions such as the distribution of twin primes or the Chebyshev bias. This algorithm can be further combined with the quantum Fourier transform to yield an estimate of the prime …

从最坏情形到平均情形再到增量搜索：强Lucas测试的界

From Worst to Average Case to Incremental Search Bounds of the Strong Lucas Test

本文研究强Lucas测试的错误界，证明通过t轮连续测试及额外标准测试的整数为素数的概率大于1-(4/15)^t，并首次给出基于强Lucas测试的增量搜索算法的错误界。

素数幂域上椭圆曲线的殆素数阶

Almost Prime Orders of Elliptic Curves Over Prime Power Fields

本文研究素数 p 使得 |E(F_{p^l})| / |E(F_p)| 具有有界素因子的无穷性，对 CM 曲线无条件应用大筛法，对非 CM 曲线在 GRH 下应用 Chebotarev 密度定理，并利用向量筛法给出相关殆素数个数的下界。

There is entanglement in the primes

Large series of prime numbers can be superposed on a single quantum register and then analyzed in full parallelism. The construction of this Prime state is efficient, as it hinges on the use of a quantum version of any efficient primality test. We show that the Prime state turns out to be very entangled as shown by the scaling properties of purity, Renyi entropy and von Neumann entropy. An analytical approximation to these measures of entanglement can be obtained from the detailed analysis of the entanglement spectrum of the Prime state, which in turn produces new insights in the Hardy-Littlew…