A-Level Maths: A1-15 Proving there are Infinitely Many Primes

[Dan-uibm]

I have realised I will never fully understand this so I'm just gonna memorise it

[andredgy]

For people that might not understand:
If P is a prime number, it is larger than the list we included to make P (which was supposed to contain all the prime numbers), so clearly there are an infinite amount of primes.
However if P is not a prime, it needs to be divisible by a number which is a prime, we know that isnt true as the remainder would always be 1, as we added 1 onto the product.

[JP-crvf]

I finally understand this!!! 4 hours before my first exam!!!! You are a

[phuongtn]

I don't quite understand the part where "P must have a prime factor which must be a factor of 1" at 4:55, can somebody explain it for me:, )?

[paulhunter]

Really helpful with the short explanation at the end. Thank you very much 👏

[Anteater]

An alternative explanation.

As each pi divides p1p2...pn then certainly pi does not divide p1p2...pn +1 since pi>1 (it will have remainder 1).

[seanpatten]

Why does the prime factor need to be a factor of 1?

[reece]

What do you mean by factor of 1 and why is it impossible?

[christophergould]

Sort of understand. Takes a bit of getting used to. The basis is that a whole number must have dividends of prime numbers stati g with two. 42has pri. É number di isors of 2, 37 and 2x3x7 equals 42.The next non prime is 44 divisible by 2.Both 42 and 44 are divisible by prime 2. and 2 extra is divisible by 2.But P +1 implies that one of the primes that helped create P must be a factor of one and that is impossible. Have I understood it?

[lukeollerhead]

Original list could be (2, 3, 5, 7, 11) multiplied out then adding 1 you get 2311. You have stated that it is sufficient to say "it will not divide by any of the numbers in the original list and it is not a factor of 1". How does that prove it does not divide by any of the primes factors that may exist between 11 and 2311? say 23 and 29.

[dansilverman]

Is P a factor of 1 because if you divided p1, p2, p3...pn by one of p1, p2, p3...pn you would get a remainder of 1 each time which would have to be divided by one of p1, p2, p3...pn?

[edwardhudson]

why must the prime factor of p be a factor of 1

[thesudaneseprince]

I do not understand with the +1 comes from...

[ianwingrove]

Surely it doesnt have to be a factor of one? It could have a factor of a prime number greater than the largest prime in the list, hence it will not go into the product of the primes but would possibly go into the product of the primes + 1

[redroses]

Hi. Say you assume there are a finite no of primes and let P be the list of all prime nos and divide any one of the primes in this list by P and get a reminder of one. why does this suggest mean that P is a prime no? As a way of proof of contradiction

[rasheede.o]

But wouldn't this always be prime- p = 2*3*5*7 + 1. I was told at school if you multiply prime numbers and then add 1 you get a prime number.

[misan]

Sir can you please explain why you add a one?

[SlahDah]

How do you not know that after multiplying all of the primes and adding one there isn’t a factor that isn’t prime

[lukas]

so there is no non prime factor of the product? As it would have to be prime ?

[regularman]

Hi,

Why can't P be a factor of 1, isn't that all numbers. I don't understand how that is impossible.