Primes with no sevens
This is a prime number whose decimal digits are all ones. It has 317 ones. It's not the world record. The number with 1031 ones is also known to be prime!
Even larger guys like this are suspected
to be prime. Are there infinitely many? Mathematicians believe so, but they can't prove it.
Why do they believe it? The main reason is that they have an estimate of the "probability" that a number with some number of digits is prime. We can use this to guess the answer to this puzzle.
Of course the whole idea of "probability" is a bit weird here. A number is either prime or not: the math gods do not flip coins to decide which numbers are prime!
Nonetheless, treating primes as if
they were random turns out to be useful. Mathematicians have made many guesses using this idea, and then proved these guesses are right, using a lot of extra work.
Of course it's subtle. If I wrote down a number with 317 twos in its decimal expansion, you'd instantly know it's not prime - because it would be even.
In the European Congress of Mathematics, a number theorist named James Maynard just announced something cool. There are infinitely many prime numbers with no sevens in their decimal expansion!
And his proof works equally well for any other number: there infinitely many primes without 0 as a digit, or 1, or 2, or 3, or 4, or 5, and so on.
This is big news, but not because mathematicians really care about primes with no sevens in them. It's because proving something like this requires a deep and delicate understanding of "the music of primes" - the way prime numbers are connected to wave patterns. For more on that, here's something easy to read:https://plus.maths.org/content/missing-7s
Thanks to +Luis Guzman
for pointing out this article, and thanks to +David Roberts
for finding James Maynard's paper on this subject, which is here:
• James Maynard, Primes with restricted digits, http://arxiv.org/abs/1604.01041
He shows that if your base b is sufficiently large, you can find infinitely many primes that are lacking a chosen set of digits, where this set can contain up to b^(23/80) of the digits. Unfortunately I don't see how large b must be - he may not have worked this out. If b = 10 counts as sufficiently large, then since 10^(23/80) is about 1.94, this result would let you avoid any one digit in base 10, but not two. In any event, he does prove, separately, that you can find infinitely many primes that avoid any one digit in base 10.
It uses cool techniques, like "decorrelating Diophantine conditions which dictate when the Fourier transform of the primes is large from digital conditions which dictate when the Fourier transform of numbers with restricted digits is large". It also uses ideas from Markov process theory - that is, the theory of random processes - as well as hard-core number theory concepts. #bigness #spnetwork
arXiv:1604.01041 #numberTheory #primes