We say that a prime number $p$ is an $\textit{Artin prime}$ for $g$ if mod generates the group $(\mathbb{Z}/p\mathbb{Z})^{\times}$. For appropriately chosen integers $d$ and $g$, we present conjecture asymptotic $\pi_{d,g}(x)$ of primes $p \leq x$ such both $p+d$ are Artin $g$. In particular, identify class pairs $(d,g)$ which $\pi_{d,g}(x) =0$. Our results suggest distribution pairs, amongst o...

Fermat’s Factoring Algorithm (FFA) is an integer factorisation methods factoring the modulus N using exhaustive search. The appearance of Estimated Prime Factor (EPF) method reduces cost FFA’s loop count. However, EPF does not work for balanced primes. This paper proposed modified 1-Estimated (mFFA1-EPF) that improves method. algorithm works a with unbalanced and primes, respectively. main resu...

A prime sieve is an algorithm that finds the primes up to a bound n. We say that a prime sieve is incremental, if it can quickly determine if n+1 is prime after having found all primes up to n. We say a sieve is compact if it uses roughly √ n space or less. In this paper we present two new results: • We describe the rolling sieve, a practical, incremental prime sieve that takes O(n log logn) ti...

Bringmann and Dousse recently established a conjecture of Dyson dealing with the limiting asymptotics of the Andrews-Garvan crank statistic for integer partitions. A direct “sieving” technique is used to establish this conjecture and establish the range of validity. Unlike the approach of Bringmann and Dousse, the technique readily yields the analogous result for Dyson’s partition rank and all ...

We present the pseudosquares prime sieve, which finds all primes up to n. Define p to be the smallest prime such that the pseudosquare Lp > n/(π(p)(logn)); here π(x) is the prime counting function. Our algorithm requires only O(π(p)n) arithmetic operations and O(π(p) log n) space. It uses the pseudosquares primality test of Lukes, Patterson, and Williams. Under the assumption of the Extended Ri...

Tate pairings has found several new applications in cryptography. However, how to compute Tate pairing is a research focus in all kinds of applications of pairing-based cryptosystems (PBC). In the paper, the structure of Miller’s algorithm is firstly analyzed, which is used to implement Tate pairing. Based on the characteristics that Miller’s algorithm will be improved tremendous if the order o...

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In this article, I seek to find whether the type of government in a country affects the amount of education provided. To do this, I propose a model where political power is distributed throughout the population, and their choice to support a government is based on an observable factor; whether the government provides access to education. Using the results from this model, I propose 3 hypotheses...

Executive Summary Ensuring safety requires reducing the risk of incidents, faults and failures that can disrupt normal operations. This effort goes far beyond simply installing fail-safe controllers or a safety instrumented system. In fact, to mitigate the risk of serious incidents that can cause injury to personnel, equipment and the environment, it is important to consider safety from all asp...