Closing the Gap: The Quest to Understand Prime Numbers by Vicky Neale

Author:Vicky Neale [Neale, Vicky]
Language: eng
Format: epub
Tags: Mathematics, General, Number Theory, History & Philosophy
ISBN: 9780191092442
Google: wpc4DwAAQBAJ
Publisher: Oxford University Press
Published: 2017-10-04T16:00:00+00:00

Returning to our hypothetical 15-sided pencil, the theorem tells us that each of the 1, 2, 4, 7, 8, 11, 13 and 14 sides contains infinitely many primes. Wow!

As we have already seen, we can prove some special cases of this one at a time. Proving the full theorem needs quite a lot more work, and I’m not going to talk you through that—it requires more mathematical technology than I want to introduce here. But I think that I can give you a flavour of the kind of argument that gets used.

Let’s go back to the primes, rather than subsets of the primes. We’ve already seen Euclid’s proof that there are infinitely many primes (suppose not, multiply them all together and add one, get a contradiction). As I described above, we can adapt that idea for some individual cases of Dirichlet’s Theorem, but each needs its own modification, and so to prove the general result we need a new idea. I’d like to outline another proof that there are infinitely many primes, since this can be adapted more conveniently for Dirichlet’s Theorem. This argument is attributed to Euler, the 18th century Swiss mathematician who’s already appeared in this book as the recipient of Goldbach’s letter containing his famous conjecture—it’s not often that Euler gets a mention only as having received a letter, since he was one of the most prolific mathematicians in history!

Here’s Euler’s idea. Take all the primes in the world, take their reciprocals (that is, take 1 divided by each of them), and add up the answers. So we add , , , , , and so on. We might write the sum as

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