
A Number Theorist Who Solves the Hardest Easy Problems - theafh
https://www.quantamagazine.org/james-maynard-solves-the-hardest-easy-math-problems-20200701/
======
magneticnorth
Now I'm curious about that last result discussed, for bases less than 10 -
"proving that there are infinitely many primes that don’t have any 7s (or any
other digit you might choose)". It sounds like Maynard decided to stop at
proving the result for all bases 10 and above, but could it be true for the
smaller bases too?

In the case of binary numbers, proving that there are infinitely many prime
binary numbers without a 0 is the same as proving that there are infinitely
many Mersenne primes, so the (non-trivial) binary case seems to still be a
difficult open problem.

It would be fascinating if there's a cutoff in base expansions where this
stops being true.

~~~
kevinventullo
I suspect in that in any base b, and for any collection of base b digits such
that the gcd of the whole set is 1, there are infinitely many primes which
only use those digits. I also suspect the Mersenne case b=2, set={1}, is the
hardest.

------
kevinventullo
This left a bit of bad taste in my mouth: “It’s easy to imagine an alternative
timeline in which Zhang proved his result six months after Maynard instead of
six months before.”

I don’t think even Maynard has claimed he would have gotten to bounded gaps
without Zhang.

~~~
alimw
The article does call Maynard's approach "completely independent". So while
Maynard himself might not want to make any such claim, clearly the impression
left with the interviewer is that he would indeed have got there without
Zhang.

------
amflare
Does this mean that for any given prime, there will always be a prime within
600 digits? Or that given a set of infinite numbers, you can also construct an
infinite set of pairs of primes 600 digits apart?

~~~
ogogmad
> Does this mean that for any given prime, there will always be a prime within
> 600 digits?

No. Euclid showed that the gap between consecutive primes can be arbitrarily
large. In particular, n!+2, n!+3, ..., n!+n are all composite whenever n>1.

> Or that given a set of infinite numbers, you can also construct an infinite
> set of pairs of primes 600 digits apart?

It means that there are infinitely many pairs of primes p and q such that 0 <
|p - q| < 600.

