Euclid's Proof: Infinity of Primes

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Euclid's proof of the infinity of primes begins with the assumption of a finite number of primes, designating the largest as Pn. By constructing a number X as the product of all primes plus one, it is shown that X cannot be evenly divided by any of the primes, as it yields a remainder of one. This leads to the conclusion that there must be primes not included in the original list, thus proving the infinity of primes. The discussion also touches on the grammatical accuracy of referring to "the infinity of primes," suggesting "the infinite set of primes" as the correct form. Overall, the proof effectively demonstrates that there are infinitely many primes through logical reasoning and division.
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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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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.
 
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".
 
kenewbie said:
How can I know this?
Try dividing.
 
HallsofIvy said:
\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.

That made perfect sense, thank you!

k
 
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