what steps need more justification or explanation
what steps need more justification or explanation
Proof:
1) Proof by contradiction, so I need to show that the negation is false.
2) Negation: There exists a largest prime number N.
3) Suppose there exists a largest prime number N. Then all numbers bigger than N can be factored into some smaller primes.
4) Consider the product N!+1= (1*2*3*4 … *N) +1
5) Any number between 1 and N will have a remainder of 1 when you try to divide it into N! +1.
6) Rephrasing: (N!+1)/x is not a whole number for any integer x in {1, … N}.
7) That means none of those numbers {1, … ,N} can be factors of N!+1 because they don’t divide in evenly.
8) So N!+1 has no factors between 1 and N. But all of the primes are between 1 and N because N is the largest prime. So N!+1 would need to be a prime!
9) But N!+1 is bigger than N. Contradiction
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