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A new class of primes?
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Mathematicians Find a New Class of Digitally Delicate Primes
https://www.quantamagazine.org/mathematiciansfindanewclassofdigitallydelicateprimes20210330
 


Mathematicians Find a New Class of Digitally Delicate Primes
https://www.quantamagazine.org/mathematiciansfindanewclassofdigitallydelicateprimes20210330
I am so glad we have Quanta Magazine. /JeppeSN  

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Joined: 16 Feb 10 Posts: 820 ID: 55391 Credit: 729,976,157 RAC: 385,487

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.  


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  

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Joined: 16 Feb 10 Posts: 820 ID: 55391 Credit: 729,976,157 RAC: 385,487

And there's your next OEIS sequence!  

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Joined: 16 Feb 10 Posts: 820 ID: 55391 Credit: 729,976,157 RAC: 385,487

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.
 

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