Wednesday, August 3, 2016

Playing With Primes...

James Maynard continues his fascinating work with prime numbers:
Maynard has shown that there are an infinite number of primes that include no number “7” digit. As the article states:
" ‘The vast majority of big numbers have lots and lots of 7s in them, so having no 7s is a rare property for any whole number to have.’ The fact that, despite this rarity, Maynard was able to prove that there are infinitely many such primes counts as an impressive feat. There's nothing special about the number 7, incidentally. Maynard's proof works equally well for any other number: so we now know that there are infinitely many primes without 1 as a digit, or 2 as a digit, or 3, or 4, or 5, and so on.”
The article proceeds with some interesting exposition about primes, music, Fourier analysis, “waves,” and cuboids.
But returning to the digit analysis I can’t help but wonder if there are also an infinite number of primes missing any two digits, say “7” and “3” for example, or “1” and “2”… or of course then three digits… or…. (what are the limiting cases here, and what, if anything, would they mean?). Or, starting at the other end, are there an infinite number of primes composed of say, nothing but 1s or 7s  (DOHH!!, that wouldn't work).
I don’t see any indication in the piece how much of this has already been looked at? HEY, Mike Lawler, a weekend project! ;-)

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