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I am having difficulty in understanding the proof of Behmann of Infinitude of primes. Can someone please explain the last part 'The proof is concluded by noticing....' which is in page $178$?

Any help would be appreciated. Thanks in advance.

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  • $\begingroup$ You have been asking many questions about different proofs that there are infinitely many primes. Are you on a mission to understand every proof that there are infinitely many primes, or are you looking up all these proofs for a broader reason? $\endgroup$
    – KCd
    Jun 20, 2020 at 18:46
  • $\begingroup$ Being on a mission is fine with me. What I don't get is how math is fun when you a) have trouble understanding each and every proof of the infinitude of primes and b) show no inclination at all at doing the thinking yourself. $\endgroup$ Jun 20, 2020 at 19:32
  • $\begingroup$ @franz ..how do you know I am having trouble understanding each and every proof..I ask only those which I don't understand.. $\endgroup$ Jun 20, 2020 at 19:56
  • $\begingroup$ Moreover, I understood the first part..I asked for the last paragraph..where did you see I asked for a whole proof? $\endgroup$ Jun 20, 2020 at 19:56
  • $\begingroup$ Yeah .I was curious to look at some of he proofs. $\endgroup$ Jun 20, 2020 at 20:00

1 Answer 1

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Suppose there are only $m$ primes. The expression

$$\frac{p_i}{p_i-1} = \frac{1}{1-\frac{1}{p_i}} = 1 + \frac{1}{p_i} + \frac{1}{p_i^2} +\cdots,$$

by geometric series.

So if the right hand side is multiplied out, you get every possible unit fraction. So the right hand side must be equal to the harmonic series. But we've just shown that the harmonic series is strictly larger. So there must be more than $m$ primes.

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  • $\begingroup$ You didn't get my question..I meant last paragraph which is in page 178 $\endgroup$ Jun 20, 2020 at 18:34
  • $\begingroup$ The last paragraph starts on page 177. Maybe you should be more clear about what your question really is. $\endgroup$
    – B. Goddard
    Jun 21, 2020 at 2:55
  • $\begingroup$ See my edit.... $\endgroup$ Jun 21, 2020 at 4:56
  • $\begingroup$ See mine....... $\endgroup$
    – B. Goddard
    Jun 21, 2020 at 13:06
  • $\begingroup$ Yeah.... I already got it... anyway..thanks.. $\endgroup$ Jun 21, 2020 at 14:19

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