
#1
Jan2110, 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 selfobvious to me. How can I know this? k 



#2
Jan2110, 06:36 AM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 38,886

Because dividing by any of those primes gives a remainder of 1, not 0:
[tex]\frac{P_1P_2...P_{i1}P_iP_{i+1}...Pn+ 1}{Pi}= P_1P_2...P_{i1}P_{i+1}...Pn+ \frac{1}{P_i}[/tex] The first term is an integer but the second is not. 



#3
Jan2110, 12:34 PM

Emeritus
Sci Advisor
PF Gold
P: 16,101

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".




#5
Jan2210, 07:54 AM

P: 236

k 


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