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# The infinity of primes

1. ### kenewbie

236
Euclid's proof:

1) Assume there is a finite number of primes.
2) Let Pn be the largest prime.
3) Let X be the P1 * P2 ... * Pn + 1

At this point the statement is that "X cannot be divided by P1 through Pn", but why is that? This is not self-obvious to me. How can I know this?

k

2. ### HallsofIvy

40,410
Staff Emeritus
Because dividing by any of those primes gives a remainder of 1, not 0:

$$\frac{P_1P_2...P_{i-1}P_iP_{i+1}...Pn+ 1}{Pi}= P_1P_2...P_{i-1}P_{i+1}...Pn+ \frac{1}{P_i}$$
The first term is an integer but the second is not.

3. ### Hurkyl

16,090
Staff Emeritus
Minor aside: "the infinity of primes" is bad grammar. The noun form is wrong here -- you want the adjective "infinite", such as in "the infinite set of primes".

4. ### Hurkyl

16,090
Staff Emeritus
Try dividing.

5. ### kenewbie

236
That made perfect sense, thank you!

k