Articles containing proofs | Prime numbers | Integer sequences

A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself.However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow method of checking the primality of a given number , called trial division, tests whether is a multiple of any integer between 2 and . Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always produces the correct answer in polynomial time but is too slow to be practical. Particularly fast methods are available for numbers of special forms, such as Mersenne numbers. As of December 2018 the largest known prime number is a Mersenne prime with 24,862,048 decimal digits. There are infinitely many primes, as demonstrated by Euclid around 300 BC. No known simple formula separates prime numbers from composite numbers. However, the distribution of primes within the natural numbers in the large can be statistically modelled. The first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability of a randomly chosen large number being prime is inversely proportional to its number of digits, that is, to its logarithm. Several historical questions regarding prime numbers are still unsolved. These include Goldbach's conjecture, that every even integer greater than 2 can be expressed as the sum of two primes, and the twin prime conjecture, that there are infinitely many pairs of primes having just one even number between them. Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers. Primes are used in several routines in information technology, such as public-key cryptography, which relies on the difficulty of factoring large numbers into their prime factors. In abstract algebra, objects that behave in a generalized way like prime numbers include prime elements and prime ideals. (Wikipedia).

Interesting Facts About the Last Digits of Prime Numbers

This video explains some interesting facts about the last digits of prime numbers.

From playlist Mathematics General Interest

Prime Factoring - GCSE Mathematics Revision (Foundation)

What are prime numbers? Learn how to find the prime factors of a number and write it as a product of prime factors. ❤️ ❤️ ❤️ Support the channel ❤️ ❤️ ❤️ https://www.youtube.com/channel/UCf89Gd0FuNUdWv8FlSS7lqQ/join

From playlist Number

Introduction to prime numbers for GCSE 9-1 maths!

From playlist Prime Numbers, HCF and LCM - GCSE 9-1 Maths

ALGEBRA & PRE-ALGEBRA REVIEW: Ch 1 (17 of 53) What Are Prime Numbers?

Visit http://ilectureonline.com for more math and science lectures! In this video I will how how to determine if numbers are prime numbers. Next video in this series can be seen at: https://youtu.be/ktUueQ8bcWI

From playlist Michel van Biezen: MATH TO KNOW BEFORE HIGH SCHOOL

Prime Numbers and their Mysterious Distribution (Prime Number Theorem)

Primes are the building blocks of math. But just how mysterious are they? Our study of prime numbers dates back to the ancient Greeks who first recognized that certain numbers can't be turned into rectangles, or that they can't be factored into any way. Over the years prime numbers have

From playlist Prime Numbers

Algebra - Ch. 6: Factoring (4 of 55) What is a Prime Number?

Visit http://ilectureonline.com for more math and science lectures! In this video I will explain what is a prime number. A prime number is a positive integer that can only be written as a product of one and itself. Its factors are “1” and itself. To donate: http://www.ilectureonline.com/

From playlist ALGEBRA CH 6 FACTORING

How to Tell if a Number is a Prime Number

This tutorial explains how to determine whether or not a number is a prime number. Join this channel to get access to perks: https://www.youtube.com/channel/UCn2SbZWi4yTkmPUj5wnbfoA/join :)

From playlist Basic Math

The Music of the Primes - Marcus du Sautoy

The Music of the Primes Marcus du Sautoy, Oxford University Thursday, May 8, 2008, at 6:00 pm MIT, Compton Laboratories Building 26, Room 26-100 Access via 60 Vassar Street Marcus du Sautoy, author of the The Music of the Primes, will discuss the mystery of prime numbers, the hi

From playlist Science

Fun with Math: Surprises with Arithmetic and Numbers

Stephen Wolfram shows kids and adults some fun unique things you can do with math. All demonstrations powered by the Wolfram Language. Originally livestreamed at: https://twitch.tv/stephen_wolfram Follow us on our official social media channels: Twitter: https://twitter.com/WolframRese

From playlist Stephen Wolfram Livestreams

CTNT 2018 - "The Biggest Known Prime Number" by Keith Conrad

This is lecture on "The Biggest Known Prime Number", by Keith Conrad, during CTNT 2018, the Connecticut Summer School in Number Theory. For more information about CTNT and other resources and notes, see https://ctnt-summer.math.uconn.edu/

From playlist CTNT 2018 - Guest Lectures

The Biggest Known Prime Number - Keith Conrad [2018]

Slides for this talk: https://ctnt-summer.math.uconn.edu/wp-content/uploads/sites/1632/2018/05/mersennetalkCTNT.pdf May 29: Keith Conrad (UConn) Title: The Biggest Known Prime Number. Abstract: There are infinitely many primes, but at any moment there is a biggest known prime. Earlier t

From playlist Number Theory

Closing the Gap: the quest to understand prime numbers - Vicky Neale

Oxford Mathematics Public Lectures: Vicky Neale - Closing the Gap: the quest to understand prime numbers Prime numbers have intrigued, inspired and infuriated mathematicians for millennia and yet mathematicians' difficulty with answering simple questions about them reveals their depth and

From playlist A Vicky Neale Playlist

Introduction to number theory lecture 1.

This lecture is the first lecture of my Berkeley math 115 course "Introduction to number theory" For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj53L8sMbzIhhXSAOpuZ1Fov8 This lecture gives a survey of some of the topics covered later in the course,

From playlist Introduction to number theory (Berkeley Math 115)

Structure and randomness in the prime numbers - Terence Tao

Speaker : Terence Tao ( Department of Mathematics, UCLA ) Venue : AG 66, TIFR, Mumbai Date and Time : 23 Feb 12, 16:00 "God may not play dice with the universe, but something strange is going on with the prime numbers" - Paul Erdos The prime numbers are a fascinating blend of both struc

From playlist Public Lectures

A History of Primes - Manindra Agrawal [2002]

2002 Annual Meeting Clay Math Institute Manindra Agrawal, American Academy of Arts and Sciences, October 2002

From playlist Number Theory

Introduction to number theory lecture 5. Primes.

This lecture is part of my Berkeley math 115 course "Introduction to number theory" For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj53L8sMbzIhhXSAOpuZ1Fov8 We discuss some basic properties of primes and prove the fundamental theorem of arithmetic.

From playlist Introduction to number theory (Berkeley Math 115)

An easy intro to prime numbers and composite numbers that MAKES SENSE. What are prime numbers? A prime number is a number that has exactly 2 factors: two and itself. What are composite numbers? A composite number is one which has two or more factors. What is the difference between a p

From playlist Indicies (Exponents) and Primes

Oxford Mathematics Public Lectures: James Maynard - Prime Time: How simple questions about prime numbers affect us all. Numbers are fascinating, crucial and ubiquitous. The trouble is, we don't know that much about them. James Maynard, one of the leading researchers in the field explains

From playlist Oxford Mathematics Public Lectures