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Message boards : General discussion : A new class of primes?

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Message 149862 - Posted: 5 Apr 2021 | 20:20:37 UTC

Mathematicians Find a New Class of Digitally Delicate Primes

https://www.quantamagazine.org/mathematicians-find-a-new-class-of-digitally-delicate-primes-20210330

Message 149863 - Posted: 5 Apr 2021 | 22:20:45 UTC - in response to Message 149862.

Mathematicians Find a New Class of Digitally Delicate Primes

https://www.quantamagazine.org/mathematicians-find-a-new-class-of-digitally-delicate-primes-20210330

I am so glad we have Quanta Magazine. /JeppeSN

Message 149864 - Posted: 5 Apr 2021 | 22:21:30 UTC - in response to Message 149862.

That article made me think of another kind of counting problem:
How many ways do you get a prime number by changing one zero bit in a particular prime number into a one?

For example:
17 in binary (most significant bit first) is 10001; when you you change that to 10011 you have 19, but that's it, any other zero bit changed to one is results in a composite number: 10101 is 21, 11001 is 25; so the answer for 17 is 1 (unless you keep going with higher powers of 2; is there a limit? 17+8192=8209 another prime). NB amazingly adding powers of 2 to 17 produces a few unexpected perfect squares: 17+8=25, 17+32=49, 17+64=81, 17+512=529

The basic problem is finding how many primes you can generate from another prime by adding powers of 2, skipping those powers of 2 that already occur in the additive decomposition into powers of 2 of the prime.

Message 149865 - Posted: 5 Apr 2021 | 22:58:47 UTC - in response to Message 149864.

That article made me think of another kind of counting problem:
How many ways do you get a prime number by changing one zero bit in a particular prime number into a one?

For example:
17 in binary (most significant bit first) is 10001; when you you change that to 10011 you have 19, but that's it, any other zero bit changed to one is results in a composite number: 10101 is 21, 11001 is 25; so the answer for 17 is 1 (unless you keep going with higher powers of 2; is there a limit? 17+8192=8209 another prime). NB amazingly adding powers of 2 to 17 produces a few unexpected perfect squares: 17+8=25, 17+32=49, 17+64=81, 17+512=529

The basic problem is finding how many primes you can generate from another prime by adding powers of 2, skipping those powers of 2 that already occur in the additive decomposition into powers of 2 of the prime.

I take it to mean that you can replace one existing zero (excluding leading zeroes, only internal zeros are allowed (Mersenne primes have no zeroes to try)) by a one. The first prime for which you can produce 2 other primes in this way, is 43. The first where you can produce 3 other primes, is 149. Continuing like that, I get:
[1, 2] [2, 43] [3, 149] [4, 4421] [5, 5441] [6, 49169] [7, 542021] [8, 2376131] [9, 19154321]
/JeppeSN

Message 149866 - Posted: 5 Apr 2021 | 23:33:05 UTC - in response to Message 149865.

And there's your next OEIS sequence!

Message 149867 - Posted: 6 Apr 2021 | 0:18:06 UTC

How about organizing sets of primes by their Hamming distance (number of bit flips).

Then questions could be asked such as, are there any finite sets of primes (a constellation? a galaxy?) whose members are reachable from at least one other member via a single bit flip (Hamming distance = 1).

The distinguishing feature of a set would be that it is separated from other sets by a minimum Hamming distance of 2. Are there any isolated primes (no neighbours with Hamming distance = 1)? If none, then these sets are a covering of the primes.

Is a covering the way to prove Goldbach's conjecture?
OK, I spilled the beans with the last question. I thought about using this approach to proving the conjecture a few years ago.

Message boards : General discussion : A new class of primes?