Numbers with a large prime factor

Numbers with a large prime factor

Large Gaps between Consecutive Prime Numbers

Let G(X) denote the size of the largest gap between consecutive primes below X . Answering a question of Erdős, we show that G(X) > f(X) logX log logX log log log logX (log log logX) , where f(X) is a function tending to infinity with X . Our proof combines existing arguments with a random construction covering a set of primes by arithmetic progressions. As such, we rely on recent work on the e...

Acceleration detection of large (probably) prime numbers

In order to avoid unnecessary applications of Miller-Rabin algorithm to the number in question, we resort to trial division by a few initial prime numbers, since such a division take less time. How far we should go with such a division is the that we are trying to answer in this paper?For the theory of the matter is fully resolved. However, that in practice we do not have much use.Therefore, we...

Odd Perfect Numbers Have a Prime Factor Exceeding

It is proved that every odd perfect number is divisible by a prime greater than 107.

Odd perfect numbers have a prime factor exceeding 108

Jenkins in 2003 showed that every odd perfect number is divisible by a prime exceeding 107. Using the properties of cyclotomic polynomials, we improve this result to show that every perfect number is divisible by a prime exceeding 108.