**Thursday 24th July**

**Number Theory**

**Matt Parker**

Twitter: http://www.twitter.com/standupmaths

Matt Parker, known as the Stand-up Mathematician, can be seen talking about Maths on the BBC, in The Guardian and on stages across the UK – at science fairs, festivals and in theatres. Once a normal Maths Teacher, his first love is still visiting schools through Think Maths to give engaging maths talks and run hands-on activities. Matt talks about Mathematics for organisations including the Royal Institution and the BBC and he was the People’s Choice Award in the 2009 national Famelab competition. He was the 2011 winner of the Joshua Phillips award for Public Engagement and he is outreach coordinator in the School of Mathematical Sciences at Queen Mary university.

Number theory, or higher arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers, sometimes called “The Queen of Mathematics” because of its foundational place in the discipline. Number theorists study prime numbers as well as the properties of objects made out of integers (e.g., rational numbers) or defined as generalizations of the integers (e.g., algebraic integers).

http://en.wikipedia.org/wiki/Number_theory

The writers of the Simpsons often hide mathematics in their episodes.

The Simpsons is an American animated sitcom created by Matt Groening for the Fox Broadcasting Company. The series is a satirical depiction of a middle class American lifestyle epitomized by the Simpson family, which consists of Homer, Marge, Bart, Lisa, and Maggie. The show is set in the fictional town of Springfield and parodies American culture, society, television, and many aspects of the human condition.

http://en.wikipedia.org/wiki/The_Simpsons

The Simpsons and Their Mathematical Secrets is a 2013 book by Simon Singh, which is based on the premise that “many of the writers of The Simpsons are deeply in love with numbers, and their ultimate desire is to drip-feed morsels of mathematics into the subconscious minds of viewers”. The book compiles all the mathematical references used throughout the show’s run and analyses them in detail. Rather than just explaining the mathematical concepts in the context of how they relate to the relevant episodes of The Simpsons, Singh “uses them as a starting point for lively discussions of mathematical topics, anecdotes and history

http://en.wikipedia.org/wiki/The_Simpsons_and_Their_Mathematical_Secrets

http://www.simonsingh.net/Simpsons_Mathematics/

https://www.youtube.com/watch?v=bk_Kjpl2AaA

In the episode “Marge and Homer Turn a couple Play (Season 17 2006) there is significant mathematical bit at the end of the episode. The numbers on the image refer to the possible attendance at a baseball game.

As you can see the numbers are 8191, 8128, 8208 (or no way to tell). Each one of those numbers is there for a special reason. They are mathematically significant.

**8191 is a Mersenne prime number **

In mathematics, a Mersenne prime is a prime number of the form

where n is also a prime number. This is to say that it is a prime number which is one less than a power of two. They are named after the French monk Marin Mersenne who studied them in the early 17th century. The first four Mersenne primes are 3, 7, 31 and 127.

http://en.wikipedia.org/wiki/Mersenne_prime

2 to the power of 13 minus 1 equals 8191

Not all Mersenne numbers are prime numbers. In 2008 a prize was given for finding the largest Marsenne prime number. It contains 17 million digits.

http://en.wikipedia.org/wiki/Great_Internet_Mersenne_Prime_Search

Prime numbers are very important for internet security

**8128 is a perfect number**

In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself (also known as its aliquot sum). Equivalently, a perfect number is a number that is half the sum of all of its positive divisors (including itself) i.e. σ_{1}(n) = 2n.

This definition is ancient, appearing as early as Euclid’s Elements (VII.22) where it is called perfect, ideal, or complete number. Euclid also proved a formation rule (IX.36) whereby p(p+1)/2 is an even perfect number whenever p is what is now called a Mersenne prime. Much later, Euler proved that all even perfect numbers are of this form. This is known as the Euclid–Euler theorem.

It is not known whether there are any odd perfect numbers, nor if infinitely many perfect numbers exist.

The first perfect number is 6. You can divide it by 1, 2 and 3 and if you add 1, 2 and 3 you get 6.

28 can be divided by 1, 2, 4, 7 and 14. If you add those numbers together you get 28.

You have to wait a long time to get to the next one which is 496 so they are not that frequent.

http://en.wikipedia.org/wiki/Perfect_number

8128 is the fourth perfect number.

Rene Descartes said that perfect number were as rare as perfect men.

http://en.wikipedia.org/wiki/Ren%C3%A9_Descartes

René Descartes (31 March 1596 – 11 February 1650) was a French philosopher, mathematician and writer who spent most of his life in the Dutch Republic.

**8208** has four digits and if you raise each digit to the fourth power and add them together you will get 8208, i.e.

The number regenerates itself from its own components.

This sort of number is called a narcissistic number

In recreational number theory, a narcissistic number (also known as a pluperfect digital invariant (PPDI), an Armstrong number (after Michael F. Armstrong) or a plus perfect number) is a number that is the sum of its own digits each raised to the power of the number of digits. This definition depends on the base b of the number system used, e.g. b = 10 for the decimal system or b = 2 for the binary system.

http://en.wikipedia.org/wiki/Narcissistic_number

These numbers are very rare and at the moment there are less than a hundred of them.

The largest one at the moment is 115,132,219,018,763,992,565,095,597,973,971,522,401

Another Narcissistic prime is 371

**More Mersenne Primes**

**Making perfect numbers from Mersenne primes**

Euclid’s use of Mersenne primes (which weren’t called that yet, since Mersenne wasn’t born until nearly 2,000 years later) was ingenious.

7 is a Mersenne prime. Multiply it by the largest power of 2 that’s less than 7, which is 4: 7 x 4 = 28, a perfect number.

The next Mersenne prime is 31. Multiply it by the previous power of 2, which is 16, and you get 496 (a perfect number too!)

http://www.mersenneprimes.com/Connection-to-Perfect-Numbers.html

http://www.math.caltech.edu/papers/Ma7_ch12.pdf

If n = 3 then the Marsenne prime is 7 and the perfect number is 28

If n = 7 then the Marsenne prime is 127 and the perfect number is 8128

All perfect numbers are the product of primes

Ri maths masterclasses:

http://www.rigb.org/education/masterclasses

Further Maths Support Programme: