Prev: A Feasible Optimal Solution to the P Versus NP Problem: l=7
Next: Skew-symmetric matrix A and I + A - Cayley transform
From: master1729 on 1 May 2010 13:19
proof of hardy-littlewood 2nd conjecture.

consider the interval (a,b).

we need to prove that the number of primes in that interval cannot be larger than pi(b - a).

thus pi(a,b) <= pi(b - a).

let the interval lenght be c = b - a.

we only need to prove it for large c , since for small it has been checked by modular arithmetic.

we need to sieve all primes up to sqrt(c).

to prove the conjecture , we need to consider the worst case scenario.

for clarity i will give examples , but it can be restated formally.

7 < sqrt(c).

the primes are the ones who are not sieved.

for every interval , 7 sieves it C or C - 1 times.

c =/= C for clarity.

thus the most times 7 can sieve , compared to the least time differs by one time , since C - ( C - 1 ) = 1.

thus an interval of length c is sieved by a prime p =< sqrt(c) at worst one time less than the interval (0,c).

thus we add +1 prime ( since not sieved ) for every p < sqrt(c).

this gives us a formula for the maximum number of unsieved numbers in the interval c , since we simply do

sum (n=p) f(n)=1 going from 2 till sqrt(c).

this is simply pi(sqrt(c)).

thus an interval c appears to have at most

pi(c) + pi(sqrt(c)) primes.

however note that pi(c) requires a correction too.

since 2,3,5,7,... are no longer counted as primes , we need to remove these.

and we need to remove up to sqrt(c) , thus we arrive at
- pi(sqrt(c))

then we end up with :

max is the same as sup.
c = b - a

max pi(a,b) = pi(c) + pi(sqrt(c)) - pi(sqrt(c))

which reduces to :

max pi(a,b) = pi(c)

thus

thus pi(a,b) <= pi(b - a).

which is equivalent to hardy-littlewood 2nd conjecture.

Q.E.D.


the master

tommy1729
From: Physicest on 1 May 2010 20:26
1) mathforum

2) idiot

"master1729" <tommy1729(a)gmail.com> wrote in message
news:1230284925.56257.1272748799416.JavaMail.root(a)gallium.mathforum.org...
> poof of hardy-littlewood 2nd conjecture.


From: spudnik on 1 May 2010 21:21
so, your coinage of pi(a,b) is the same as pi(b) - pi(a); now,
can you say thr proof as a wordprolemmum?

--Light: A History!
http://wlym.takeTHEgoogolOUT.com
From: christian.bau on 2 May 2010 11:08
On May 1, 10:19 pm, master1729 <tommy1...(a)gmail.com> wrote:
> proof of hardy-littlewood 2nd conjecture.
>
> consider the interval (a,b).
>
> we need to prove that the number of primes in that interval cannot be larger than pi(b - a).
>
> thus pi(a,b) <= pi(b - a).

Please post your proof again after adding a definition of pi (a, b).
From: master1729 on 4 May 2010 06:51
gnasher729 wrote :

> On May 1, 10:19 pm, master1729 <tommy1...(a)gmail.com>
> wrote:
> > proof of hardy-littlewood 2nd conjecture.
> >
> > consider the interval (a,b).
> >
> > we need to prove that the number of primes in that
> interval cannot be larger than pi(b - a).
> >
> > thus pi(a,b) <= pi(b - a).
>
> Please post your proof again after adding a
> definition of pi (a, b).

pi(a,b) is the number of primes in the interval (a,b).
 |  Next  |  Last
Pages: 1 2 3 4 5 6
Prev: A Feasible Optimal Solution to the P Versus NP Problem: l=7
Next: Skew-symmetric matrix A and I + A - Cayley transform