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Mathematicians hunting prime numbers discover infinite new pattern

scientificamerican.com

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Re: Mathematicians hunting prime numbers discover infinite new pattern

#3

Post by _feus »

wewewedxfgdf wrote: post #2

This sort of thing makes me feel there is some deep understanding of reality only inches away from us, we glimpse it through these patterns but the secret remains hidden.

and it will be something so trivial and obvious, those who were looking for it will be kicking themselves for missing it

Re: Mathematicians hunting prime numbers discover infinite new pattern

#5

Post by waltbosz »

wewewedxfgdf wrote: post #2

This sort of thing makes me feel there is some deep understanding of reality only inches away from us, we glimpse it through these patterns but the secret remains hidden.

Wouldn't it be fun if someone out there already knows a simple way to determine if a number is prime without factoring, but to them it is so obvious that they didn't even consider others may be interested.

Re: Mathematicians hunting prime numbers discover infinite new pattern

#7

Post by Sniffnoy »

I'm a little confused at the significance here. Before I read the definition of the M_a, this seemed crazy, but on actually reading it, M_1 is just the sum-of-divisors function (usually denoted sigma).

So, n is prime iff M_1(n)=n+1. That's much simpler than the first equation listed there!

Indeed, looking things up, it seems that in general the functions M_a can be written as a linear combination (note: with polynomial coefficients, not constant) of the sigma_k (sigma_k is the sum of the k'th power of the divisors). So this result becomes a lot less surprising once you know that...

Re: Mathematicians hunting prime numbers discover infinite new pattern

#8

Post by seanmcdirmid »

_feus wrote: post #3
wewewedxfgdf wrote: post #2

This sort of thing makes me feel there is some deep understanding of reality only inches away from us, we glimpse it through these patterns but the secret remains hidden.

and it will be something so trivial and obvious, those who were looking for it will be kicking themselves for missing it

It’s a huge refrain that shows up again every 20 years or so. Wolfram wrote a huge book with this premise, but I don’t think it’s gone anywhere even though it’s surely 25 years old by now.

Re: Mathematicians hunting prime numbers discover infinite new pattern

#9

Post by vasvir »

waltbosz wrote: post #5
wewewedxfgdf wrote: post #2

This sort of thing makes me feel there is some deep understanding of reality only inches away from us, we glimpse it through these patterns but the secret remains hidden.

Wouldn't it be fun if someone out there already knows a simple way to determine if a number is prime without factoring, but to them it is so obvious that they didn't even consider others may be interested.

Well I have a really elegant proof for this but I don't have enough space in the HN reply box to write it out -- but it is trivial, I am sure you will work it out.

   Fermat Reincarnation.

Re: Mathematicians hunting prime numbers discover infinite new pattern

#10

Post by bubblyworld »

Sniffnoy wrote: post #7

I'm a little confused at the significance here. Before I read the definition of the M_a, this seemed crazy, but on actually reading it, M_1 is just the sum-of-divisors function (usually denoted sigma). So, n is prime iff M_1(n)=n+1. That's much simpler than the first equation listed there! Indeed, looking things up, it seems that in general the functions M_a can be written as a linear combination (note: with polynomi…

Can you elaborate? How does this result become less surprising if you know that? Personally I would not have guessed that there are infinitely many characterisations of P involving sums-of-powers-of-divisors either.
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