What caused Euclid and Euler to prove this theorem?
What is the significance of infinite number of primes?
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prime-numbers
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3Infinitude of primes is less trivial than infinitude of composite numbers. In general, all infinitude/finitude results are of interest, unless they are really trivial. – 2017-02-04
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1maybe because if the primes were finite it would mean that some integers cannot be produced by multiplication. – 2017-02-04
1 Answers
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Because it is by no means trivial that there are infinitely many prime numbers, to begin with. If you consider the first prime numbers 2, 3, 5, 7, 11, 13, etc you can see that they occur frequently, but as you move to much higher integers, you realize that they become less nad less frequent. So it is not at all obvious that there are infinitely many of them and a theorem stating their cardinality is needed.