Number Theory: Queen of Mathematics

Mathematician Sarah Hart will be giving a series of lectures on Maths and Money.

--------------------------------

In which years does February have five Sundays? How many right-angled triangles with whole-number sides have a side of length 29? How many shuffles are needed to restore the order of the cards in a pack with two Jokers? Are any of the numbers 11, 111, 1111, 11111, . . . perfect squares? Can one construct a regular polygon with 100 sides if measuring is forbidden? How do prime numbers help to keep our credit cards secure?

These are all questions in number theory, the branch of mathematics that’s primarily concerned with our counting numbers, 1, 2, 3, etc. Of particular importance are the prime numbers, the ‘building blocks’ of our number system.

The subject is an old one, dating back to the ancient Greeks, and for many years has been studied for its intrinsic beauty and elegance, not least because several of its challenges are so easy to state that everyone can understand them, and yet no-one has ever been able to resolve them.

This lecture situates the above problems and puzzles in their historical context, drawing on the work of many of the greatest mathematicians of the past, such as Euclid, Fermat, Euler and Gauss. Indeed, as Gauss, sometimes described as the ‘Prince of Mathematics’, has claimed: Mathematics is the Queen of the Sciences, and Number Theory is the Queen of Mathematics.

A lecture by Robin Wilson

The transcript and downloadable versions of the lecture are available from the Gresham College website:

[PopeLando]

43:00 calculating the day of the week, I love this stuff.
As well as memorising the number you add for each month, you might as well memorise the first part as 1900s=0, 1800s=2, 1700s=4 and 1600s (and 2000s)=6, and the cycle repeats every four centuries.
I don't quite understand this complicated rule for the year part. Just add the year (ie 20) to the number of leap years (5) for the same result mod 7.

[markkennedy]

This is a great talk. So accessible yet fascinating.

[robharwood]

Excellent use of simpler, more everyday-type of examples (e.g. clock/week arithmetic, shuffling cards) to give concrete examples of much more abstract/complex application (e.g. general modular arithmetic, 'shuffling' our credit details).

[hannamakela]

Very Short Introduction is such a brilliant publication series!!!

[vijaysingbundhoo]

When we talk of Number Theory The names of G H Hardy FRS and S Ramanujan FRS need to be mentioned and the legacy they have left in that field whereas the recent one being Paul Erdos the Hungarian Mathematician.

[abcde_fz]

I love finding these quirks:
~ 16:20
In the section on "perfect numbers", the example graphic lists the number:
33, 550, 366
But in the example proof section, the 'proof equation' uses the number:
33, 550, 336

Which is correct?

[ValMartinIreland]

Taking a deck of cards with no jokers, well shuffled between each examination with the aim of getting the exact same arrangement as on starting out. So starting out kc, 6h, Jd, 4c, As, 10d and so on. The goal is to get back to kc, 6h, Jd, 4c, As and 10 d and so on. The formula is 52 X 51X 50X 49 X 48 X 47 X 46 and so on. No ordinary computer ot calculator could show such an amorous number.

[frejahedeman]

I LOVE it! Thank you for making maths so exciting! Or how to understand a lot of maths in 1 go.

[FreemanPresson]

"Prime figures of my story" : I see what you did there.

[roberthumphreys]

Gresham lectures are always so worthwhile!

[Legendfound-xe]

Extremely interesting. Thank you so much, Sir.

[venkatbabu]

Prime means clusters not breakable and other things can be broken. What does that mean. Distribution can occur only in specific pattern dependent on breakage. Construction is stable with prime coefficients. Others vibrate. Usually combination has special vibration equations. Chemical reactions occur when there is a juggler. Each level depends on atomic or subatomic or other groups. To get everything use 1/12 th limiting coefficients. Use π within a spectrum. Light is supposed to have 12. 6 white 6 dark one carry over to 7. 5 unknown through fingers.

[badhombre]

While you're still processing the last thing said, he looks up with that expression of "more?", and instantly the answer is yes.
Brilliant presentation.

[Carvin]

Very nice! I ordered the book, and the one on combinatorics too! These are pedagogical gems.

[kds]

Yes indeed and RSA is also used in securing logons for systems using the Public/Private key exchange. Now if only I'd learned this in my early schooling math would have made much more sense. Of course studying Point Set Topology made sense too because it has in common with computing.

[sciencefordreamers]

Thank you so very much Professor, great work, very professional!

[robcampbell]

That was a fascinating Mathematical story, thanks.

[jimvj]

The proof of Fermat's little theorem by counting necklaces was (first?) given by Solomon Golomb in 1956 (see Wiki).
I wonder why he did not give him credit.

[davidcovington]

Guess I'll list all those ways of finding primes and then study at it for fun. Thank you for such a clear presentation.

[craffte]

Great basics, great pacing. Thank you very much. Number theory tastes like music.