The infinity of primes


by kenewbie
Tags: euclid proof primes
kenewbie
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#1
Jan21-10, 06:27 AM
P: 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
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HallsofIvy
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#2
Jan21-10, 06:36 AM
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Because dividing by any of those primes gives a remainder of 1, not 0:

[tex]\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}[/tex]
The first term is an integer but the second is not.
Hurkyl
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#3
Jan21-10, 12:34 PM
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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".

Hurkyl
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#4
Jan21-10, 12:35 PM
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The infinity of primes


Quote Quote by kenewbie View Post
How can I know this?
Try dividing.
kenewbie
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#5
Jan22-10, 07:54 AM
P: 236
Quote Quote by HallsofIvy View Post
[tex]\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}[/tex]
The first term is an integer but the second is not.
That made perfect sense, thank you!

k


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