STRUCTURE AND RANDOMNESS IN THE PRIME NUMBERS 1
... pseudorandomly in many ways, and not to follow any simple pattern. We have many ways of establishing that a pattern exists... but how does one demonstrate the absence of a pattern? In this article I will try to convince you why the primes are believed to behave pseudorandomly, and how one could try ...
... pseudorandomly in many ways, and not to follow any simple pattern. We have many ways of establishing that a pattern exists... but how does one demonstrate the absence of a pattern? In this article I will try to convince you why the primes are believed to behave pseudorandomly, and how one could try ...
On the Probability of Relative Primality in the Gaussian Integers
... Number theory is historically defined as the study of the integers and is one of the oldest fields of mathematics to be developed with sophistication. Throughout antiquity, multiple mutually exclusive civilizations developed basic mathematical notions regarding shape and number. By the 2nd century B ...
... Number theory is historically defined as the study of the integers and is one of the oldest fields of mathematics to be developed with sophistication. Throughout antiquity, multiple mutually exclusive civilizations developed basic mathematical notions regarding shape and number. By the 2nd century B ...
On consecutive integers
... prime numbers . Thus we can assume n > 2k. b) Assume first 2k < n < k3'`2 . By (*) there are least k primes amongst the integers (9), but since ...
... prime numbers . Thus we can assume n > 2k. b) Assume first 2k < n < k3'`2 . By (*) there are least k primes amongst the integers (9), but since ...
factoring, prime numbers and prime factorization
... Prime Numbers If a number does not have a factor besides 1 and itself, it is called prime. For example, 5 has no factors, whole numbers that can be divided into it evenly, besides 1 and 5. It is therefore prime. 20, on the other hand, has the factors 1, 2, 4, 5, 10 and 20. Because it has factors bes ...
... Prime Numbers If a number does not have a factor besides 1 and itself, it is called prime. For example, 5 has no factors, whole numbers that can be divided into it evenly, besides 1 and 5. It is therefore prime. 20, on the other hand, has the factors 1, 2, 4, 5, 10 and 20. Because it has factors bes ...
![THE RING Z[ √ D] - facstaff.bucknell.edu](http://s1.studyres.com/store/data/013114635_1-8d703d2c2f33989febabe1cd6c40a070-300x300.png)
THE RING Z[ √ D] - facstaff.bucknell.edu
... Consider 2 for example. If 2 = αβ in Z[ −5] then taking norms gives 4 = N (α)N (β). We cannot have N (α) = 2 since one can easily check that the equation a2 + 5b2 = 2 has no integer solutions. This forces one of N (α), N (β) to equal 1 in which case the corresponding element is a unit. This proves t ...
... Consider 2 for example. If 2 = αβ in Z[ −5] then taking norms gives 4 = N (α)N (β). We cannot have N (α) = 2 since one can easily check that the equation a2 + 5b2 = 2 has no integer solutions. This forces one of N (α), N (β) to equal 1 in which case the corresponding element is a unit. This proves t ...
