Infinite Primes - Numberphile

How do we know there are an infinite number of primes?
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Dr James Grime explains, with a bit of help from Euclid.

NUMBERPHILE

Videos by Brady Haran

[SomeRandomFellow]

5:50 this man just divided a line into 7ths first try

[NXaiUL]

4:47 "Mathematics is the search for truth and this book is as true today as it was 2300 years ago" - James Grime

[anosmianAcrimony]

5:44 Probably the most impressive thing he does in this video

[mrembeh]

He can divide a line in perfect seven parts

[Relative]

4:49: "Mathematics is the search for Truth" - Absolutely! That is why I became a mathematician!

[KT-djiy]

What puzzled me for ages about Euclid’s proof was what Grimes says at 1:43 and, especially, at 2:09. What I didn’t at the time realize was that proof calls on another famous piece of maths, the _Fundamental Theorem of Arithmetic, _ from which we can be confident that every integer greater than 1 is either prime, or is the product of primes where that product is unique. It’s part of the beauty of maths that I was able to be impressed at Euclid’s proof without knowing (or even knowing _about)_ the _Fundamental Theorem of Arithmetic._ But still, it niggled me, and eventually when I learned about it I was even more impressed. 🙂

[titotitoburg]

I have a question:

Suppose the only primes are 2, 3, 5, 7, 11, 13 and 17. And I want to construct a number Q by multiplying all of them together, and then adding 1.

2 x 3 x 5 x 7 x 11 x 13 x 17 = 510, 510 +1 = 510, 511 which is not a prime.


So... what gives? Who's to say Q in your proof isn't like 510, 511?


edit: did some thinking and 510, 511 is factored by 19, 97, and 277, none of those being on the list. therefore the list was not complete. Well played mathematics... you win again.

[owenpeter]

Never mind the Math, where can I get the brown paper roll?

[varAstorsson]

Grime´s enthusiasm for mathematics makes me enthusiastic about mathematics!

[stevennunn]

Thanks for your reference to Euclid's Elements, after watching this video, I purchased the book and it's brilliant.

[rachelbrain]

I love Dr. Grime's enthusiasm for his field of study. ^^

[KevinVanOrd]

Seeing the passion you guys have for numbers and mathematics is inspiring. It shows me that what is mundane to you could be incredibly meaningful to me, and that no intellectual pursuit is without value. I don't have the same passion for numbers, but I do have passion for other things, and the fun and knowledge I gain by watching these encourages me to go and exercise my own talents and intellect. I am grateful for that.

[pigman]

5:38 he draws a nearly perfectly straight line and then divides it into seven nearly perfectly equal parts like it's no big deal?!?!

[gfetco]

Translations:
Ashume -> Assume
Noombar -> Number
Drawring -> Drawing

[medhatmostafa]

I actually like euclid's definition better than modern days

[GordonjSmith]

Loved this, a specific book reference re: Euclids Elements would be great!

[russellthorburn]

6:03 All these years and I've never heard of a geometric definition of a prime before. Very nice.

[moonlightviolinist]

Thank you for pointing that out to me. I guess I hadn't thought about the fact that since the list is by definition incomplete, it doesn't have to be the set of all prime numbers within a given range, just that the answer will never be divisible by any number in the series. If you leave 2 out of the series, your product will be divisible by 2 (and not any of the numbers in your series), proving that you have an incomplete list. Makes sense now.

[zilvoxidgod]

It took me a couple rewatches to understand that proof, but now that I do, I see why you called it elegant!

[johnny]

I was doing some math and found that (2n)+(n^2)-1 created primes very well if n is even. Example: (2 x is prime. I also saw that up to 200 being n (leaving out odd numbers) it spit out a prime 42% of the time.