Primality Testing

2016 ◽  
pp. 283-314
Keyword(s):  
Primality Testing

scholarly journals Dreibens Modulo 7: A New Formula for Primality Testing

Elements ◽  
2016 ◽  
Vol 12 (1) ◽  
Author(s):  
Arthur Diep-Nguyen
Keyword(s):  
Number Theory ◽  
Linear Algebra ◽  
Abstract Algebra ◽  
Additional Insight ◽  
Primality Testing ◽  
New Formula ◽  
Insight Into ◽  
Rule Out

In this paper, we discuss strings of 3’s and 7’s, hereby dubbed “dreibens.” As a first step towards determining whether the set of prime dreibens is infinite, we examine the properties of dreibens when divided by 7. by determining the divisibility of a dreiben by 7, we can rule out some composite dreibens in the search for prime dreibens. We are concerned with the number of dreibens of length n that leave a remainder i when divided by 7. By using number theory, linear algebra, and abstract algebra, we arrive at a formula that tells us how many dreibens of length n are divisible by 7. We also find a way to determine the number of dreibens of length n that leave a remainder i when divided by 7. Further investigation from a combinatorial perspective provides additional insight into the properties of dreibens when divided by 7. Overall, this paper helps characterize dreibens, opens up more paths of inquiry into the nature of dreibens, and rules out some composite dreibens from a prime dreiben search.

scholarly journals An open source software package for primality testing of numbers of the form p2^n+1, with no constraints on the relative sizes of p and 2^n

2018 ◽  
Author(s):  
Tejas R. Rao
Keyword(s):  
Open Source ◽  
Open Source Software ◽  
Software Package ◽  
Primality Testing ◽  
Fermat Number ◽  
Open Source Software Package ◽  
Sierpinski Numbers ◽  
Efficient Software

We develop an efficient software package to test for the primality of p2^n+1, p prime and p>2^n. This aids in the determination of large, non-Sierpinski numbers p, for prime p, and in cryptography. It furthermore uniquely allows for the computation of the smallest n such that p2^n+1 is prime when p is large. We compute primes of this form for the first one million primes p and find four primes of the form above 1000 digits. The software may also be used to test whether p2^n+1 divides a generalized fermat number base 3.

scholarly journals A REMARK ON PRIMALITY TESTING AND DECIMAL EXPANSIONS

2011 ◽  
Vol 91 (3) ◽  
pp. 405-413 ◽  
Author(s):  
TERENCE TAO
Keyword(s):  
Upper Bound ◽  
Primality Testing ◽  
Positive Proportion ◽  
Fixed Base ◽  
Covering Set ◽  
Selberg Sieve

AbstractWe show that for any fixed base a, a positive proportion of primes become composite after any one of their digits in the base a expansion is altered; the case where a=2 has already been established by Cohen and Selfridge [‘Not every number is the sum or difference of two prime powers’, Math. Comput.29 (1975), 79–81] and Sun [‘On integers not of the form ±pa±qb’, Proc. Amer. Math. Soc.128 (2000), 997–1002], using some covering congruence ideas of Erdős. Our method is slightly different, using a partially covering set of congruences followed by an application of the Selberg sieve upper bound. As a consequence, it is not always possible to test whether a number is prime from its base a expansion without reading all of its digits. We also present some slight generalisations of these results.

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