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I'm confused by the "Admissible Combs" sidebar. Can someone help me understand? "Roughly speaking, a comb is admissible if there is no obvious reason why its t
by abuehrle 13y ago
I'm confused by the "Admissible Combs" sidebar. Can someone help me understand?
"Roughly speaking, a comb is admissible if there is no obvious reason why its teeth couldn’t point entirely to primes infinitely often as you move it along the number line"
Then: "A much more audacious conjecture called the prime k-tuple conjecture ... posits that any admissible comb will point entirely to primes infinitely often."
Isn't this just saying the prime-tuple conjecture states that admissible combs are admissible?
- justinpombrio 13y agoAdmissibility doesn't say that a comb's teeth will point entirely to primes infinitely often, it merely says (as you quoted) that there's no obvious reason its teeth won't point entirely to primes infinitely often. I suspect that "obvious reason" is being used as shorthand for some technical definition the article doesn't try to state. Any mathematician here that knows more?
- impendia 13y agoSure. {n, n + 2, n + 4} is a 3-tuple that is not admissible: One of n, n + 2, and n + 4 will always be divisible by three, so the tuple has no chance to ever represent three prime numbers for one value of n. (Except for n = 3.) Conversely, {n, n + 2, n + 6} is an admissible 3-tuple. There's no fixed integer that will always divide at least one of n, n + 2, and n + 6, so we believe that there are infinitely many n for which n, n + 2, and n + 6 are all prime.
- justinpombrio 13y agoAnd so the claim is that there is no less obvious reason that a comb might not be all primes only finitely often. That is an audacious claim :-).