yet another 'proof' of the legendary conjecture

[guptadeva]

http://citeseerx.ist.psu.edu/viewdoc...=rep1&type=pdf

the common denominator of many recent 'proofs' floating around on the internet seem to be an inadequate use of the english language ... but ok, let the doubt be in favor of the 'accused'

the pdf cited above becomes slightly more readable when substituting the word 'term' or 'expression' instead of the word 'equation'

other 'proofs' are either cyclic (the author assumes the validity of the conjecture in order to prove the conjecture) or simply blurred by chaotic or unrelated sets of equations.

however ... it is always interesting to see which tools or ideas are being presented in these 'proofs' - and since probably nobody today is honestly doubting the validity of the conjecture itself (?) it would be nice to have a real proof someday !

so ... what's the catch here ?

[CRGreathouse]

The first mistake I see is in (4) on page 2. floor((n+1)^2/m) - floor(n^2/m) is indeed at least floor((2n+1)/m), but that doesn't mean you can make the substitution -- you need to majorize/round *up* on terms you're subtracting (and minorize/round down on terms you're adding) if you want to prove a lower bound. Otherwise, just note that floor((n+1)^2/m) - floor(n^2/m) >= 0 and "conclude" that (3) is at least 2n+1.

This is a major, fundamental mistake at the heart of the proof, so it certainly doesn't hold.

[guptadeva]

This is a major, fundamental mistake at the heart of the proof, so it certainly doesn't hold.

exactly

somehow it's also good to see that the number of correct and insightful papers on this subject is > 0 like e.g. in the following example:

https://arxiv.org/pdf/1310.1323.pdf

[science_man_88]

Originally Posted by guptadeva

exactly

somehow it's also good to see that the number of correct and insightful papers on this subject is > 0 like e.g. in the following example:

https://arxiv.org/pdf/1310.1323.pdf

Many things can be interconnected. The twin prime conjecture, can be a statement about goldbach partitions. Goldbach's conjecture, restated as an equidistance conjecture on most of the natural number line.

[CRGreathouse]

Originally Posted by guptadeva

exactly

somehow it's also good to see that the number of correct and insightful papers on this subject is > 0 like e.g. in the following example:

https://arxiv.org/pdf/1310.1323.pdf

The result in Remark 1 on Conjecture 1 (p. 3) could be improved using work carried out on this forum.

[guptadeva]

ok ... yet another fresh and original one:

https://justmathstuff.wordpress.com/...s-conjectures/

the author uses pi(x) to mean the conventinal pi(x)-1 not counting 2 as a prime

some statements are not correct as e.g. in the case x=4:

the pattern 3 a b 3 leading to the conclusion that 3 primes are needed to 'accomodate' a target set of four elements is wrong.

counter-example: the target set 11 13 15 17 needs 4 primes (11,13,3,17) to be 'accomodated'

hence without additional combinatorical arguments, the lemma and the main result are not proven ...

also the statement of the idea: 'each new prime discovered by the sieve is nothing more than the next number that the number 3 failed to accommodate' ... would lead to 35 being prime number

[science_man_88]

Originally Posted by guptadeva

ok ... yet another fresh and original one:

https://justmathstuff.wordpress.com/...s-conjectures/

the author uses pi(x) to mean the conventinal pi(x)-1 not counting 2 as a prime

some statements are not correct as e.g. in the case x=4:

the pattern 3 a b 3 leading to the conclusion that 3 primes are needed to 'accomodate' a target set of four elements is wrong.

counter-example: the target set 11 13 15 17 needs 4 primes (11,13,3,17) to be 'accomodated'

hence without additional combinatorical arguments, the lemma and the main result are not proven ...

also the statement of the idea: 'each new prime discovered by the sieve is nothing more than the next number that the number 3 failed to accommodate' ... would lead to 35 being prime number

Ah, but for 4 consecutive natural numbers it is true it only takes 3 apply the pigeonhole principle. Okay it relies on positioning properly.

[guptadeva]

Originally Posted by science_man_88

Okay it relies on positioning properly.

as much as i like the idea of counting the number of times some multiples of each prime p_i with i<=pi(n) falls into the interval [(n-1)^2 , n^2] - in order to actually give a proof of the legendre conjecture there is still an 'epsilon' missing in the argument ... any ideas how to fill the gap ?

[robtaylor501]

Originally Posted by guptadeva

ok ... yet another fresh and original one:

https://justmathstuff.wordpress.com/...s-conjectures/

the author uses pi(x) to mean the conventinal pi(x)-1 not counting 2 as a prime

some statements are not correct as e.g. in the case x=4:

the pattern 3 a b 3 leading to the conclusion that 3 primes are needed to 'accomodate' a target set of four elements is wrong.

counter-example: the target set 11 13 15 17 needs 4 primes (11,13,3,17) to be 'accomodated'

hence without additional combinatorical arguments, the lemma and the main result are not proven ...

also the statement of the idea: 'each new prime discovered by the sieve is nothing more than the next number that the number 3 failed to accommodate' ... would lead to 35 being prime number

Hello,

I noticed that you're citing my work...but coming to faulty conclusions. Your alleged "counter-example" is not a counter-example of my proof at all. The LEAST number of primes needed to accommodate a target set of four elements is indeed pi(4)+2=3. This is what my proof is concerned with: it answers the question regarding the LEAST number of primes needed. It is possible to find a set of four consecutive odd numbers requiring MORE than 3 primes to accommodate. Indeed, you found such a set with 11 13 15 17. Yet you will never find a set of four consecutive odd numbers that can be accommodated with LESS than 3 primes. Get it?

As for your statement on my discussion regarding the sieve, (a) it has nothing to do with my proof, as my discussion on the sieve was simply providing some background on the history of prime numbers; and (b) I feel you're being a bit too nitpicky. Obviously, what I was saying is that the FIRST odd number not captured by the sieve after running it with 3 (i.e. 5) becomes our next prime, and 5 then becomes the next number we use to run the sieve, etc. We all know how the sieve works. It's elementary. Attempting to use my short background discussion on it in an effort to discredit the remainder of my work is not productive.

Okay, that's the end of my rant. At the end of the day, I am THRILLED people are actually thinking about my work, and I am more than happy to address any questions, comments, concerns, etc. you have regarding my proofs of Legendre's, Brocard's, and Andrica's, as I am certain that my proofs are correct. I love math, I don't bite, and I'm a lawyer, so I believe my grasp of the English language is more than adequate.

Let's discuss!

Cheers,

Rob Taylor, Esq.

[chalsall]

Originally Posted by robtaylor501

I love math, I don't bite, and I'm a lawyer, so I believe my grasp of the English language is more than adequate.

Let's discuss!

Let's.

Do you prefer to code in C, C++, Perl, Python, PHP, Go or C#?

[robtaylor501]

Originally Posted by chalsall

Let's.

Do you prefer to code in C, C++, Perl, Python, PHP, Go or C#?

That assumes I know how to code. I don't.