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From: Harald Helfgott on 14 Jul 2010 11:26
I can't be the only person here who has fond memories of the problems
in Vinogradov's Elements of Number Theory. (For people who have not
read it - the text itself is just a concise basic number theory book;
most of the substance is in the problems.) At the same time, much of
what is good about them became clear to me only later. Vinogradov does
not give citations or credits (not even to himself!), let alone any
sort of historical overview for why the ideas he introduces in the
problem sets are important. Standard names for the theorems the
student is asked to prove are also completely missing.

Wouldn't it be a good idea to craft a commentary on Vinogradov's
problems? Has anything of the sort been already done?

(Incidentally, something learned there recently made its way to
http://polymathprojects.files.wordpress.com/2010/07/polymath1.pdf)
From: Ludovicus on 15 Jul 2010 06:25
On 14 jul, 11:26, Harald Helfgott <harald.helfg...(a)gmail.com> wrote:
> I can't be the only person here who has fond memories of the problems
> in Vinogradov's Elements of Number Theory. (For people who have not
> read it - the text itself is just a concise basic number theory book;
> most of the substance is in the problems.) At the same time, much of
> what is good about them became clear to me only later. Vinogradov does
> not give citations or credits (not even to himself!), let alone any
> sort of historical overview for why the ideas he introduces in the
> problem sets are important. Standard names for the theorems the
> student is asked to prove are also completely missing.
>
> Wouldn't it be a good idea to craft a commentary on Vinogradov's
> problems? Has anything of the sort been already done?
>
> (Incidentally, something learned there recently made its way tohttp://polymathprojects.files.wordpress.com/2010/07/polymath1.pdf)

Your Polymath's citation is a superseded question. The AKS algorithm
determine primality in (log n)^k operations, [Much less than n^(1/2)]
From: Harald Helfgott on 15 Jul 2010 08:30

> Your Polymath's citation is a superseded question. The AKS algorithm
> determine primality in (log n)^k operations, [Much less than n^(1/2)]

Not so. The fact that the AKS algorithm determines primality in (log
n)^k operations doesn't mean
that the AKS algorithm can *find* a prime between N and 2N in (log
n)^k operations.
From: Ludovicus on 15 Jul 2010 18:38
On 15 jul, 08:30, Harald Helfgott <harald.helfg...(a)gmail.com> wrote:
> > Your Polymath's citation is a superseded question. The AKS algorithm
> > determine primality in (log n)^k ophenerations, [Much less than n^(1/2)]
>
> Not so. The fact that the AKS algorithm determines primality in (log
> n)^k operations doesn't mean
> that the AKS algorithm can *find* a prime between N and 2N in (log
> n)^k operations.

Why not?
First you form the number N0 = 6*floor[N/6],then N1 = 6*N0 + 1
Then produce the sequence U(n)= U(n)+ 4 ---> U(n+1)= U(n)+ 2
beginig with U(0) = N1. Before U(n) attains N1 + (Log N)^2
that is in [Log N)^2]/3 steps ,you will have a prime number
ready for applying AKS.
From: Gerry Myerson on 15 Jul 2010 19:32
In article
<7edc39d3-56fa-4273-bb68-9759fdb5f98d(a)5g2000yqz.googlegroups.com>,
Ludovicus <luiroto(a)yahoo.com> wrote:

> On 15 jul, 08:30, Harald Helfgott <harald.helfg...(a)gmail.com> wrote:
> > > Your Polymath's citation is a superseded question. The AKS algorithm
> > > determine primality in (log n)^k ophenerations, [Much less than n^(1/2)]
> >
> > Not so. The fact that the AKS algorithm determines primality in (log
> > n)^k operations doesn't mean
> > that the AKS algorithm can *find* a prime between N and 2N in (log
> > n)^k operations.
>
> Why not?
> First you form the number N0 = 6*floor[N/6],then N1 = 6*N0 + 1
> Then produce the sequence U(n)= U(n)+ 4 ---> U(n+1)= U(n)+ 2
> beginig with U(0) = N1. Before U(n) attains N1 + (Log N)^2
> that is in [Log N)^2]/3 steps ,you will have a prime number
> ready for applying AKS.

You seem to be suggesting that there's always a prime
between n and n + (log n)^2. A lot of people would be
interested in seeing a proof of such an allegation.

--
Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
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