Decoding some prime number conjectures

[bonzion]

1. THERE EXISTS AT LEAST TWO PRIMES NUMBERS BETWEEN N^2 AND (N+1)^2, WHERE N IS A NATURAL NUMBER.

2. THERE EXISTS AT LEAST ONE TWIN-PRIME PAIR BETWEEN N^2 AND (N+2)^2, WHERE N IS AN ODD NATURAL NUMBER.
(THEREFORE THE TWIN PRIME CONJECTURE IS TRUE)
anyone will a powerful system can verify them.

Below is a link to the first of three papers aimed at possibly settling some prime number conjectures.

In this article, we present a simple approach to tackling most prime number conjectures.

The main statement is presented as a conjecture.

http://addo.esmartdesign.com/3e.htm"

CONSTRUCTIVE COMMENTS ARE WELCOME.

 

[CRGreathouse]

The link to the Arenstorf paper is broken; the correct URL is http://arxiv.org/abs/math/0405509/. The paper was withdrawn by its author: "A serious error has been found in the paper, specifically, Lemma 8 is incorrect."

Your definition of $\Psi_m$ doesn't appear to be well-formed; I'm sure you have something particular in mind, but I can't get through the peculiar use of symbols. You seem to mean something like $\Psi_m:\mathbb{N}^m\to\mathbb{N}$ with
$$\Psi_m(x_1,\ldots,x_m)=\prod_{i=0}^mf_i(x_1,\ldots,x_m)+\underline{\alpha}(x_1,\ldots,x_m)=m\alpha+\prod_{i=0}^mf_i(x_1,\ldots,x_m)$$
but the functions f are
$$f_i(x)=a_ix+b_i$$
for $a_i\in\{1,2,\ldots\},b_i\in\{0,1,2,\ldots\}$, but then you're using a one-dimensional function as an m-dimensional function.

I don't understand the purpose of your class of functions $\underline{x}(a)$ which ignore their all of their arguments. These are constants (nullary functions) more than projection functions.

Little stuff: "Twim Prime Conjecture" should be "Twin Prime Conjecture" (p. 1). The TeX for quotes is `quotation', not 'quotation'.

 

[bonzion]

The link to the Arenstorf paper is broken; the correct URL is http://arxiv.org/abs/math/0405509/. The paper was withdrawn by its author: "A serious error has been found in the paper, specifically, Lemma 8 is incorrect."

Your definition of $\Psi_m$ doesn't appear to be well-formed; I'm sure you have something particular in mind, but I can't get through the peculiar use of symbols. You seem to mean something like $\Psi_m:\mathbb{N}^m\to\mathbb{N}$ with
$$\Psi_m(x_1,\ldots,x_m)=\prod_{i=0}^mf_i(x_1,\ldots,x_m)+\underline{\alpha}(x_1,\ldots,x_m)=m\alpha+\prod_{i=0}^mf_i(x_1,\ldots,x_m)$$
but the functions f are
$$f_i(x)=a_ix+b_i$$
for $a_i\in\{1,2,\ldots\},b_i\in\{0,1,2,\ldots\}$, but then you're using a one-dimensional function as an m-dimensional function.

I don't understand the purpose of your class of projection functions $\underline{x}(a)$ which ignore their arguments.

Little stuff: "Twim Prime Conjecture" should be "Twin Prime Conjecture" (p. 1). The TeX for quotes is `quotation', not 'quotation'.

THANKS FOR POINTING THAT OUT! IT WAS SIMPLY A TYPO. THE CORRECTION HAS BEEN MADE. THANKS

 

[CRGreathouse]

THANKS FOR POINTING THAT OUT! IT WAS SIMPLY A TYPO. THE CORRECTION HAS BEEN MADE. THANKS

What was a typo? How are you defining Phi now?

 

[dodo]

Just a little comment: In case I of theorem 4.1, why N(f) is empty when a=1? In page 6, the second example, h(x)=x+6, has N(h) = {1,2,3,4,5,6}.

 

[bonzion]

Just a little comment: In case I of theorem 4.1, why N(f) is empty when a=1? In page 6, the second example, h(x)=x+6, has N(h) = {1,2,3,4,5,6}.

N(h) = {1,2,3,4,5,6} since M(h)={7,8,9,10,11,...}

 

[dodo]

Right. Then why N(f) is empty when a=1, on case I of theorem 4.1?

h(x)=x+6 is an example of a function where a=1, yet N(f) is not empty.

 

[bonzion]

Right. Then why N(f) is empty when a=1, on case I of theorem 4.1?

h(x)=x+6 is an example of a function where a=1, yet N(f) is not empty.

HI THERE, THERE WAS A LINK TO TWO ARTICLES. PLEASE TRY AGAIN. I GUESS YOU CLICKED BEFORE THE CHANGE. THANKS

 

[dodo]

Thanks, now I see the change.

I have another comment. Please consider that I am a pregrad student, and that I find very nice what you're trying to do: you sieve the integers using a family of functions, and then try to deduce properties of the sieved set out of properties of the sieving functions. This is so cool, IMO. All the more reason to demand precision in the argument.

Theorem 4.1, cases I-III, show some function families where N(f) is finite, yet this is presented as an "if and only if", and claimed that for all other functions N(f) is infinite, without further proof.

Personally I don't find that so obvious. Look, for example, at all the effort put to show that there are infinite prime numbers. Does the proof that, for all other functions, N(f) is infinite, follow a similar pattern? Can you give a hint? Thanks.