Q: What is the prime fartorization list?

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The proof that there is no largest prime:Assume that there are a finite number of primes for the sake of contradiction. Then, there should be a number P that equals p1p2p3...pn+1. P is either prime or not prime (composite). If it is prime, we just show that P is larger than the largest prime in the list. If it's not prime, it must be composite. Composite always has at least one factor that is prime, but since P is not divisible by any prime in the list, the unknown prime factor(s) must be something not in the list, this also shows that there is a prime larger than the largest prime in the list. Both cases show that no matter how large a list of prime numbers, there will be always at least one larger prime outside of that list.

1, 31 (31 is a prime!)

23 29.

79 83 89 97.

They are 23 and 29.

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That's an unfortunate term. The answer to the prime factorization of 56 is 2 x 2 x 2 x 7.

2 x 2 x 2 x 5 = 40

474 237,2 79,3,2

It is a list of prime numbers.

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The list of prime numbers is infinite. There are two factors for each.

The list is far too long to list here (millions of them); a search on "list of prime numbers" can quickly give you the desired prime numbers.

The proof that there is no largest prime:Assume that there are a finite number of primes for the sake of contradiction. Then, there should be a number P that equals p1p2p3...pn+1. P is either prime or not prime (composite). If it is prime, we just show that P is larger than the largest prime in the list. If it's not prime, it must be composite. Composite always has at least one factor that is prime, but since P is not divisible by any prime in the list, the unknown prime factor(s) must be something not in the list, this also shows that there is a prime larger than the largest prime in the list. Both cases show that no matter how large a list of prime numbers, there will be always at least one larger prime outside of that list.

That's an infinite list.

list the prime numbers between one and twenty

It is impossible to list all the prime numbers as there are an infinite number of them. The list of prime numbers starts: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ... Suppose you could list all the prime numbers and have done so - there is no prime number which is not on the list. Multiply them all together and add 1. This number is not divisible by any of the listed prime numbers as there is always a remainder of 1. So either this new number is a prime number, or there is another prime number not listed which will divide into it. Which means that the list was not all the prime numbers; meaning it is impossible to list all the prime numbers.

That's an infinite list.