# Are There Infinitely Many Primes

@article{Goldston2007AreTI, title={Are There Infinitely Many Primes}, author={D. A. Goldston}, journal={arXiv: Number Theory}, year={2007} }

This paper is based on a talk given to motivated high school (and younger) students at a BAMA (Bay Area Math Adventure) event. Some of the methods used to study primes and twin primes are introduced.

## Figures from this paper

## 4 Citations

Elementary Primes Counting Methods

- Mathematics
- 2012

This work proposes elementary proofs of several related primes counting problems, based on an elementary weighted sieve. The subsets of primes considered here are the followings: the subset of twin…

Proof of the Twin Prime Conjecture

- Mathematics
- 2017

I can proof that there are infinitely many twin primes. The twin prime counting function π2(n), which gives the number of twin primes less than or equal to n for any natural number n, is for limn→∞…

Properties and applications of the prime detecting function: infinitude of twin primes, asymptotic law of distribution of prime pairs differing by an even number

- Mathematics
- 2011

The prime detecting function (PDF) approach can be effective instrument in the investigation of numbers. The PDF is constructed by recurrence sequence - each successive prime adds a sieving factor in…

The origin of the logarithmic integral in the prime number theorem

- Mathematics
- 2013

We establish why li(x) outperforms x/log x as an estimate for the prime counting function pi(x). The result follows from subdividing the natural numbers into the intervals s_k :={p_k^2,...,…

## References

SHOWING 1-10 OF 18 REFERENCES

Small differences between prime numbers

- MathematicsProceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences
- 1966

Let pn denote the nth prime number. The present investigation relates to the existence of relatively small values of pn+1─ pn when n is large, and establishes more precise results than those…

Primes in tuples I

- Mathematics
- 2009

We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the…

Small gaps between primes exist

- Mathematics
- 2005

In the recent preprint (3), Goldston, Pintz, and Yoldorom established, among other things, (0) liminf n→∞ pn+1 − pn log pn = 0, with pn the nth prime. In the present article, which is essentially…

SMALL GAPS BETWEEN PRIME NUMBERS : THE WORK OF

- Mathematics
- 2000

In early 2005, Dan Goldston, Janos Pintz, and Cem Yildirim [12] made a spectacular breakthrough in the study of prime numbers. Resolving a long-standing open problem, they proved that there are…

An introduction to the theory of numbers (5th edition) , by I. Niven, H. S. Zuckerman and H. L. Montgomery. Pp 529. £14·50. 1991. ISBN 0-471-5460031 (Wiley)

- Mathematics
- 1991

Theorem, that all abelian extensions of Q are subfields of cyclotomic fields. Takagi's first work of major significance, which included a new definition of class field, was to prove Kronecker's…

The Theory of the Riemann Zeta-Function

- Mathematics
- 1987

The Riemann zeta-function embodies both additive and multiplicative structures in a single function, making it our most important tool in the study of prime numbers. This volume studies all aspects…

An Introduction to the Theory of Numbers

- Philosophy
- 1938

This is the fifth edition of a work (first published in 1938) which has become the standard introduction to the subject. The book has grown out of lectures delivered by the authors at Oxford,…

Sieve Methods

- Mathematics
- 2001

Preface Sieve methods have had a long and fruitful history. The sieve of Eratosthenes (around 3rd century B.C.) was a device to generate prime numbers. Later Legendre used it in his studies of the…