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From: Archimedes Plutonium on 17 Jul 2010 04:56
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.


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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