Infinitely Many Primes--Series on Infinity Part 4

Over two thousand years ago Euclid proved that there are infinitely many prime numbers. This ingenious proof is an early example of an indirect proof--contradicting the opposite statement of a theorem. It is also a very early mathematical confrontation with infinity, in a mathematically primitive time when even algebra was unknown.

Category:

Education

Tags:

• math
• Euclid
• prime
• number
• proof
• infinity
• series
• infinitesimal

[HarryHyper]

what is infinity devided by infinity?

[Davidson1956]

In calculus we call this an indeterminate form. Zero divided by zero is also an indeterminate form, an important one at the heart of differential calculus.

There is no way to determine a numerical result for an indeterminate form, but it is usually possible to make sense of it when approaching such a form. For example, (2x^2 + 3x - 4) / (x^2 - 1) is an indeterminate form when x is infinite, however as you let x grow large towards infinity the value of this fraction approaches 2.

[yw00000]

Would this method work if you assumed 5 was the biggest prime number? 5! = 120 ... +1 =121 Which is certainly not prime. Is there something I am doing incorrectly?

[Davidson1956]

No, you're correct! This new number is not a prime number, but it is a product of prime numbers greater than 5, namely 11 times 11. And that's the idea. If P is the largest prime number, then P! + 1 is either prime or a number which contains a larger prime than P. Either situation contradicts an assumption that P is the largest prime number.

[theImaginarySolution]

Thank You. I watched this video a long time ago and a couple weeks after that we had a quizbowl tournament. One of the questions was "Euclid proved there are infinitely many of them-" then I buzzed in and said "Prime Numbers" and I got it right! And it's all thanks to this vid.

[nowenterpsie]

Again, get your precepts out for all to see:

Mathematicians such as this seems to treat Infinity as a 'number', and this is the basic and fundamental flaw in all of this logic. Infinity is a non-number, closely related to Zero, or Nothing. By definition a 'number' is bounded by finite-ness.

We are waking up.

[newagecozaki]

i feel guilty about not being constructive by watching this

[Nasbin]

I'm reading Godel, Escher, Bach and this helped a lot, thanks :)

[Crazyegg1234]

Wow! I never knew philip seymour hoffman was so good at math!

[tommyrjensen]

When we focus on some given prime number p, then by Eulid's argument, we can produce a larger one. If we suppose that p is a largest prime, then we can obtain a contradiction. Either way we may prove that the givem p is not the largest prime, and this implies that there are infinitely many primes. These two arguments are not logically quite the same. One is "direct", the other is "indirect". In the video the distinction does not become really clear.

[orangegold1]

@HarryHyper All real numbers.

[miamareye]

@SSCCMath142 thank you, but i am not really satisfied with this kind of explanation, as it is based on merely probability.

and yes, factorials get large very fast, but we know at the same time that primes tend to rarefy.

what about that?

[miamareye]

@SSCCMath142 thank you. i thougt there is a proof already.

[SSCCMath142]

I'm the guy in the video who doesn't know what he is blabbing about. Multiplying consecutive integers instead of only primes does nothing to invalidate the proof, but it does make the proof less technical and easier for many to understand.

[SSCCMath142]

Factorials get so large so fast that i would expect that some numbers of the form p! + 1 are a multiple of 3 or more primes. I can't easily put my hands on an example, but i can't imagine it not happening.

No problem with the multiple loads. If I could ever figure how to get back on Davidson1956 I'll delete the extras, and may add a few more videos.

[johndoe121213]

the guy in the video does't know what he is blabbing about. You only multiply a series of prime numbers ONLY.he multiplys primes and composites.