Number Theory: Queen of Mathematics
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- Опубліковано 20 лип 2024
- Mathematician Sarah Hart will be giving a series of lectures on Maths and Money.
Register to watch her lectures here: www.gresham.ac.uk/watch-now/s...
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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:
www.gresham.ac.uk/lectures-and...
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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.
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).
I LOVE it! Thank you for making maths so exciting! Or how to understand a lot of maths in 1 go.
This is a great talk. So accessible yet fascinating.
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.
Very Short Introduction is such a brilliant publication series!!!
My 3 am thoughs brought me here
4:39 AM
Honestly I absolutely love these lectures
12:45
What were they?
5am
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.
"Prime figures of my story" : I see what you did there.
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.
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?
33,550,336 is a perfect number, which comes from p = 13. So the corresponding Mersenne Prime is 2^13 - 1 = 8192 - 1 = 8191, and 2^12 * 8191 = 33,550,336. Search for 'List of Mersenne primes and perfect numbers' on Wikipedia and you'll see a list of all 51 currently known Mersenne Primes and their corresponding even perfect numbers.
Extremely interesting. Thank you so much, Sir.
Excellent lecture! Thanks!
Guess I'll list all those ways of finding primes and then study at it for fun. Thank you for such a clear presentation.
Fantastic lecture
Very nice! I ordered the book, and the one on combinatorics too! These are pedagogical gems.
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.
Sir it was a brilliant lecture. Also, the conclusion statement made me laugh uncontrollably😂😂
Group Theory is the queen of mathematics. Number theory is just an application of a deeper result.
Excellent!
Thank you so very much Professor, great work, very professional!
Great talk, I’m fascinated now
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.
Carry on this great work on providing free content 😇.
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.
Fermat proved his little theorem, through a simple approach. The necklace counting method was much after
Great basics, great pacing. Thank you very much. Number theory tastes like music.
35:05 no one ever taught me this until you, thanks professor
Fascinating! :)
A fine introduction to a deep topic.
Gresham lectures are always so worthwhile!
i have to disagree robert,
this was singlehandedllly the worst video i have ever watched on the internet
i have watch3ed over 25000 videos online including videos of torture and pure racism, however this video is by far the worst of them all
i really hope this changes your mind.
@@ha7vey433 why ? Please explain
@@ha7vey433 shut up
The king of number theory is Indian mathematician Srinivasa Ramanujan
nah, it is Pierre de Fermat
@@timetraveller2818 Why don't you ask all the mathematicians of the world ? Pierre de Fermat is nothing in front of Ramanujan, you have no idea about Srinivasa Ramanujan.
@@imtiazmohammad9548 ik he is a great Indian mathematician but he was doing more research about infinite series and continued fractions and elliptic functions . on the other hand pierre de fermat was more focused on number theory and made contributions like: Fermat numbers and Fermat primes, Fermat's principle, Fermat's Little Theorem, and Fermat's Last Theorem all about number theory. not saying one is better than the other.
@@timetraveller2818 Theory of partition is a great example of Srinivasa Ramanujan's contribution to number theory, who could have guessed that there will be a formula for infinite partitions. For 100 years this was a problem, Ramanujan came up with this crazy formula.
since i do not want to continue this dispute lets just end this with Fermat = Ramanujan
At 15:35 33,550,366 should read 33,550,,336 (as correctly shown further down on the same page).
even better would be: 3355,0336,,Fin. as sqrt1000 is unlike sqrt100 or sqrt10000 which are 10 and 100 respectively and so on. would make an easier numbers-notation. easier to remember Big numbers, I would say. Also p.e. this prime: (2^8258,9933Fin)-1
@@konradcomrade4845 Yes, the notation would be better, but at least we don't have the Indian system of grouping digits in sequences of unequal length!
That was a fascinating Mathematical story, thanks.
i think we all know Rob, that was a lie. the matheematical questions that this gentalman pronounced was so boring that your comment left me speechless
have a terrrible day rob
you thoughitly deserve it
@@ha7vey433 Cheers mate, you too.
Good session sir! I learnt a lot from Vidya Guru sessions as well. They post all exam relevant content.
@@ha7vey433 He was just saying a Thanks. You act like you got up on the wrong side of the manger this morning.
Thankyou for the overview.
P = NP is modular form = 8 = 4 x 2 = x^2+y^2=z^2 = 2 (mod 8)
On 9:16 minute, there are two statements: "There are infinitely many primes of the form 4n+3", and "There are infinitely many primes of the form 4n+1". I know how to prove the first one (easy) but can anybody help me with the proof of the second one?
I thought he was a real mathematician, until at the end when he made a joke that make me laugh, and then I knew he was an imposter, and not a real mathematician.
Fascinating talk, thanks.
Glad he said "sometimes called the positive integers" and not "sometimes called the natural numbers". Yup that's right, N starts at 0
ok *n e r d*
Actually, N often has different interpretations, especially in different fields of maths. It can start at 1 or at 0, and in most textbooks it’ll usually be specified. I personally use N with a subscript 1 or 0 to emphasize which one I’m using, though I think we should use the terms “positive integers” and “non-negative integers” for speech.
Edit: plus both are used equally as much. In fact, N_1 was the original N. However, constructions of the natural numbers tend to include 0.
@@integralboi2900 ik but axiomatic set theory ftw.
Calling them positive integers is less ambiguous which is bonus
@@domc3743but sowing discord tho
thank you for your persistence every digit matters
Beautiful!❤ Thank you very much!
Thanks!
Thank you
Awesome lecture professor I loved it🙏🙏🙏🙏🙏🙏🙏
Thank you for this. I think I see what math is for now.
Number theory modular form = 8 (mod n) mode of secret.
Thanks
Great!
thanks
can we draw a regular polygon with 360 sides? (as the 36 is on the list of not possible)
if I remember my class from last semester correctly we can construct a regular polygon with n sides iff n = 2^e * p_1 * p_2 * p_3 *...
for some n and with the p's pairwise different Fermat primes (meaning the p's are a prime numbers of the form 2^(2^r) +1 )
so no we can't construct a regular polygon with 360 sides since 360 = 2^3 * 3^2 *5
@@nono-mu9rw thank You. The problem is, I still don't grasp it completely. What is 2^e=?6.580885991...?
@@konradcomrade4845 e is just some natural number here, sorry that's just how our professor wrote it down in his notes😅
Another problem with mathematics is that there is no such thing in nature as minus. Minus simply means being on the opposite side of a line. East and west would do the same thing. The square root of a square plot of land comprising
1,000 meters = 31.645 (plus 31.645). The square root of a plot of land comprising minus 1000 sq meters = minus 31.645 meters. The square root of minus 1 = -1. We are told it cannot be calculated except by using i That is a myth.
Take introductory group theory and ring theory.
Thank you sir but I think one is fundamentally different from primes became it is a perfect square and primes are not perfect squares
thanks!!!!!!!!!!
Really amazing. Love from india
Sir I have observed that : Except 3 all prime numbers do not have their digital root ( sum of digits ) 3 or 6 or 9 . Is it a coincidence or some thing else ? DrRahul
If the sum of the digits is 3 or 6 or 9 it will be a multiple of 3 hence composite, the only exception being 3 itself.
@@pichaivanchinathan6527 All the prime numbers except 3 have digital root 1 or 2 or 4 or 8 or 5 or 7 .Thus there are six class of prime numbers.
I can count in tenties and I can count in up to 6 polynomial dimensions with 6 * 6 dimensional series.
Not sure quite what you're saying. Could you maybe elucidate this particular skill with an example or twotie?
@@muttleycrew tenties.
1 2 3 4 5 6 7 8 9 tenty 11....
19 tenteen 21
90 tenty..
tentytenty then 111
1 2 3 4 5 6
1 3 6 10 15 21
1 4 10 20 35 56
1 5 15 35 70 126
1 6 21 56 126 252
1 7 28 84 210 462
Now for counting with 6 * 6 dimensional series.
{1,1,1,1,1,1},{1,2,1,1,1,1},{2,1,1,1,1,1},{1,1,1,2,1,1},{1,1,2,1,1,1},(1,1,1,1,1,2),{1,1,1,1,2,1},{1,3,1,1,1,1},{2,2,1,1,1,1),
{3,1,1,1,1,1},{1,1,1,3,1,1},{1,1,2,2,1,1},
{1,1,3,1,1,1},{1,1,1,1,1,3},{1,1,1,1,2,2},{1,1,1,1,3,1},{1,2,1,2,1,1},{2,1,1,2,1,1},{1,2,2,1,1,1},{2,1,2,1,1,1},{1,2,1,1,1,2},{2,1,1,1,1,2},{1,1,1,2,1,2},{1,1,2,1,1,2},{1,2,1,1,2,1},{2,1,1,1,2,1},{1,1,1,2,2,1},{1,1,2,1,2,1},{1,4,1,1,1,1}....
one can get every number by multiplying primes? how would one get the primes themselves? and especially how can one get one?
The answer to your question is literally part of the Problems of the Millennium and for each one you solve you win a medal and a million dollars.
@@DruiB0y a million!?!
wow!
but.. bu... but that's not a prime!
@@DruiB0y no the answer is that you can't get primes and one. You can get anything else
A composite number is made up of two or more primes, which of course are not composite
(.75*(x^2))+(1.5*x)+23 = mostly prime or semi-prime numbers when x is an even number, very dense.
@@jerryiuliano871 not always? for x = 46,48,56......... it's not prime??
I don't agree that 1 is not a prime, any number that cant be expressed as repeated sum of any other number except 1 is a prime.
so 1 is a prime. you cant base definition of prime on property of multiplication which is a higher level construct of summation.
1 can be divided by itself infinite times doesnt follow the same patters as the 'real primes', 1 and 2 are often referred to as subprime
I like his first slide is some "prime" figures lol.
By 8 mins in , I was completely lost . I was probably lost right when we got to primes in the beginning. I feel so inadequate 😔
How do you write 1 as the product of primes if it is not prime itself?
You don't. 1 is not prime.
THIS COMPACTIFIES MY DIMENSION IN 3-SPACE FOR NOW.
😂
Urantia: 15:14.9 Your planet is a member of an enormous cosmos; you belong to a well-nigh infinite family of worlds, but your sphere is just as precisely administered and just as lovingly fostered as if it were the only inhabited world in all existence.
I literally wish I could have a professor like him at my college 😥😥 the algebra professor thinks we studied everything in school and teaches so freaking fast 😫😭😭
How come we can't replace the classic p = np problems by making them all square based problems using all of these classical work? If p != square or triangle or hexidecimal related algorithm then = np? If we have all the theorems in a theorem, picture a stack of algorithms like pancakes on a plate, then isn't that the same as p = np or something? I guess it matters how you look at the plate of pancakes, bottom, top, eye level, 3.14159. mmmm pancakes.
Number Theory - King of Mathematics.
Since the muses were seen, since ancient times, to be the sources of Inspiration in all things cultural, be it art or science, it would be the Queen of Mathematics.
@@Redrogue4711
Mathematics as an art and science predates the specific Greek term for it.
Along a similar vane, the oldest named attribution for the 勾股定理 ("Pythagorean" Theorem) that I know of in the entirety of World History is toward our Sage King Yu the Great 🙏姒文命大禹🙏 around 4046-4144 years ago, itself remarked on by our Royal Scholar 🙏商高🙏 3021-3121 years ago.
This predates (-570:-495) Πυθαγόρας, (-800:-600) बौधायन, and (-1800) 𒌈𒁲𒎌.
"The list of (..formation..) goes on forever" is the Prime Observation deduced from Euler's Intuitions of e-Pi-i.., of WYSIWYG here-now-forever continuous cause-effect creation event..., which for the practice of Mathematics, is THE working Theory (?), aka holographic Quantum Operator Fields Modulation Mechanism numberness quantization is this Holographic Temporal Singularity, usually represented in Polar-Cartesian self-defining infinitesimal coordination-identification positioning by logarithmic condensation module-ation.
(A real Mathematician needs to state the Proof-disproof format, Formal Reasoning Methodology, in/by the "always show your working" rule)
Logarithmic Temporal Actuality, because it's 1-0Duration density-intensity probability positioning, necessarily forms numberness dominance sequences that are inherently Quantum Computational Communication, AM-FModules, and the formulae of Chemistry that makes Number Actuality a real-time logarithmic resonance approximation in Condensed Matter. Ie occurring probabilisticly in/of phase-locked conglomerations of temporal hyper-hypo Superspin, logarithmic time-timing fluid, e and Pi sync-duration connectivity instantaneously @.dt zero-infinity i-reflection, is axial-tangential sync-duration orthogonality, Eternity-now Interval.
(The Observed Math-Phys-Chem and Geometry spin-spiral superposition, physical manifestation and field function development)
"It's more convenient to write zero.." at the Completeness of circularity instantaneous positioning of 12, ie zero-infinity sync-duration connectivity superposition.
Satisfying summary of interesting aspects of Number Theory in Actuality. Thanks
Incredibly helpful comment.
-2Pi
seems astounding
👍👍👍👍👍👍
Robin Wilson is the son of the late PM, Lord (Harold) Wilson.
O! Its oiler I thought it is euler
What if thousands are multiple of itself at a rapid state then primes are ineffective
wut
How can a polygon of 32 sides be drawn, 32 = 2^5, and there is no Fermat prime as a factor of 32, and only those polygons can be drawn which are a power of 2 x unequal Fermat primes
We can draw a 4-sided polygon (square), therefore we can do an 8, 16, and 32-sided polygon
Take a square, truncate the corners-> octagon
Take an octagon, truncate the corners -> 16-gon
Take a 16-gon, truncate the corners -> ???? profit?
0 2 3 1
3 1 0 2
1 3 2 0
2 0 1 3
Is this considered part of Number Theory?
It is a 4x4 self-orthogonal Latin square.
Whetted my appetite -- not trivial but easy to follow for the beginner -- Better than Arthur Conan Doyle!
The definition of a perfect number seems to me to be imperfect.
It could be defined in an infinite number of other ways.
Deserved many more laughs for that last joke
Followed your advice, lost all my friends. Thanks! [joking of course...]
thank you Robin I enjoyed your presentation very much
I am very sorry dudes, I'm on medication!
Sir I am from Pulwama district of Kashmir and I developed a technique for making divisibility tests of any number.
Write it down and send it to gresham college uk
I feel like I'm a wizard
How poor am I for learning nothing, sorry about it but I appreciate your effort sir the problem is my mindset
Keep going and you'll eventually understand it 👍 it's ok to not get something just believe in yourself and keep being interested
gresham
Robin ka hood nahi MILA.
۲'۳-۴-۵ و ۴۲۱_۳
In Non-Cantorian set theory, with its infinite sentences & transfinite fractions, there are numbers that negate finitudes and numbers that negate differential infinitudes - i.e. - numbers which negate identities for the non-identical [das Nichtidentische]. This exposes mathematical foundations as mortal appearance.
Four of the "Prime" figures. I see what you did there, you sneaky devil, you... 🤣😂🤦♂️👍
it was all going so well until that bloke cantor turned up
And who is the king
260722
I wonder what my former teachers would think of me, i always said id never use and want to use maths, programming and co and now all i do the entire day is solve coding challenges with math background, watching maths lecture and studying cs...
At 6:51 1001 is also a prime after 997.
1001 = 7 x 143
I'm sorry for those who do not think this is exciting 🎉
I DO
Discrete math was never my thing..
Very unfortunate Ramanujan was not mentioned. If it is not deliberate, the speaker has missed a lot in the research.
Ramanujan is not the only mathematician existing
Euler, Fermat,Euclid have even more discoveries than him
Ey, look! Another Ramanujan comment. Classic. 😒
و۶
,
MATH IS THE SCIENCE OF PATTERNS OR PERHAPS IT IS BEST TO SAY IT IS THE LANGUAGE OF PATTERNS.
If you're gonna math-major,
this is under-grad. Get ready !
My mathematics major contained no number theory whatsoever.
@@muttleycrew Booooo!!!
@@staggeredpotato6941 Okaaaay. Are you booing a mathematics degree in general or that fact that my mathematics degree didn't have any number theory or do you just dislike cartoon dogs?
@@muttleycrewI don't even know how you could do that. The first proofs class you take is typically number theory before you do group theory...
unless you are an applied math major. In that case, you're in the wrong hood, boy.
This is NOT number theory. It’s a talk about certain kinds of number …
That's what number theory is.
@@jakethemistakeRulez ," everyone said in unison.