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From: Archimedes Plutonium on 17 Jul 2010 06:40


Archimedes Plutonium wrote:
> Here is how Wikipedia states the Legendre Conjecture:
> --- quoting Wikipedia ---
> Legendre's conjecture, proposed by Adrien-Marie Legendre, states that
> there is a prime number between n^2 and (n + 1)^2 for every positive
> integer n. The conjecture is one of Landau's problems (1912) and
> unproven as of 2010.
> --- end quoting ---
>
> Here is my proof that Mersenne primes are infinite, after I doctored
> up some bad numbering.
>
>
> (1) Definition of prime as a positive integer divisible
> only by itself and 1.
>
>
> (2) The prime numbers are the numbers 2,3,5,7,11, ..,pn,... of set S
> Reason: definition of primes
>
>
> (3) The Mersenne primes are of form (2^p) -1 and the first four are 3,
> 7, 31, 127
>
>
> (4) Suppose Mersenne Primes and regular primes are finite, then
> 2,3,5,7, ..,p_n is the complete series set of Mersenne primes along
> with all the regular primes below p_n with p_n the largest Mersenne
> prime Reason: this is the supposition step
>
>
> (4.1) Set S are the only primes that exist Reason: from step (4.0)
>
>
> (4.2) Form W-1 = (2x3x5x, ..,xpn) - 1. Reason: can always operate and
> form a new number
>
>
> (4.3) Divide W-1 successively by each prime of
>  2,3,5,7,11,..pn and they all leave a remainder of 1.
> Reason: unique prime factorization theorem
>
>
> (4.4) W-1 is necessarily prime. Reason: definition of prime, step
> (1).
>
>
> (4.5) Initial cases of Mathematical Induction
>
>
> The first few Mersenne primes are 3,7,31, 127
>
>
> So the initial case of a Math Induction works for Euclid's Number as
> W +1
> {2,3} are all the primes that exist yields Euclid Number (2x3)+1 = 7
> {2,3,5} are all the primes that exist yields Euclid Number (2x3x5)+1 =
> 31
>
>
> (4.6) Assume true for case N of Mathematical Induction:
> assume true that the Euclid Number in step (4.2) of W-1 is of the form
> (2^p)-1
> and this further means that the Euclid Number of W-1 above means the
> series multiplication of (2x3x5x, ..,xpn) has the form of a number in
> the
> set (2^p) where p is prime.
>
>
> (4.7) Now must show true for Math Induction of N+1.
>
>
> (4.8) Include W-1 above into the new extended series set of
> {2,3,5,7, p_n, W-1} and translate into a new Euclid Number Y-1
> as this (2x3x5x, ..,xp_n x (W-1)) -1. And due to the Mathematical
> Induction assume N true step of (4.6) that 2x3x5x, ..,xp_n is of form
> (2^p) of
> a number
> in this series 2,4,8,16,32,.... that the number W-1 is also
> decomposable as
> that of W = (2^p) so that we have ( 2^p)^2 (-1)
>
>
> In step (4.8) I decompose the series into that of (2^p)(2^p) -1
>
>
> Step (4.9) The square of a number in the series 2,4,8,16,32, ...
> is also a member of that series
>
>
> So finally in the step (5) the Mathematical Induction of show that
> p_N
> +1 is
> satisfied is true since Y-1 is that of the form (2^p)(2^p) -1
>
>
> (5.1) Y-1 is necessarily a new prime number because all the primes
> that exist
> when divided into Y-1 leave a remainder
> (5.2) Y-1 is a Mersenne prime because of Math-Induction steps
> (5.3) Mersenne Primes are infinite because of the contradiction to
> the
> supposition
> that W-1 and then Y-1 were the last and largest Mersenne primes
> since
> the Indirect
> method reiterates another Mersenne Prime.
> (6) Mersenne Primes are infinite
> (7) Perfect Numbers are infinite
>
>
> QED
>
> Notice that to prove Mersenne primes infinite is a weaving together of
> the Indirect Euclid
> IP proof along with a intricate web of weaving the Mathematical
> Induction rule. The IP
> Indirect yields the infinitude, and the Math Induction yields or
> preserves the identity of the
> prime as a Mersenne prime.
>
> So to prove Legendre Conjecture I need to insert n^2 and then (n+1)^2
> as Euclid Numbers
> for the Indirect IP.
>
> However, I think I have a shortcut, in that the Indirect method allows
> me to insert (n-1)^2
> that is in between the Euclid Number for n^2 and (n+1)^2 and this
> Euclid Number, sandwiched
> in between the Euclid Numbers of n^2 and (n+1)^2 is necessarily prime.
> So I do not need a
> mathematical-induction steps to prove Legendre conjecture. I simply
> inject (n-1)^2 and yield that new prime number that is between n^2 and
> (n+1)^2
>
> You see, there is a new day abloom in mathematics of a new technique
> that solves most of the questions of whether a prime set is finite or
> infinite, and when someone has grabbed a hold of this new technique,
> well, I can almost clear out all these unsolved conjectures one by one
> and in short order. A new tool in engineering or math, is highly
> effective.
>

Sorry, I have a mistake in the above with the Euclid Number in the
Indirect Method
gives two new primes and that should be n^2 -1 and n^2+1 as two new
Euclid Number
primes, where we need no Math Induction. Then we do it for (n+1)^2
where we achieve
two new Euclid Number primes of (n+1)^2 +1 and (n+1)^2 -1. It is this
(n+1)^2 -1 that is
in between n^2 and (n+1)^2 of Euclid Numbers that we achieve a proof
of the Legendre
conjecture, all from Euclid Infinitude of Primes proof Indirect.

Now also note that the new prime of form n^2+1 also achieves that new
prime in between
n^2 and (n+1)^2.

So the Legendre conjecture is a darling little proof that is very easy
to prove. When we have
the proper tool- Indirect IP, some conjectures fall so easily.

Like I said before, in the World War 1 and 2 theaters, the pattern of
the Marines in combat was to get a tiny foot hold or toehold on a
beach, and from that toehold eventually you overturn the enemy. In
mathematics, a tiny overlooked mistake in the Indirect Euclid
Infinitude of Primes delivers two new primes on any supposed primes
finite list. That overlooked mistake maybe the only way of penetrating
proofs of Twin Primes, Polignac,
Mersenne, and now Legendre primes.

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
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