From: Archimedes Plutonium on 13 Aug 2010 14:58 Even geniuses slip. If I had it to do over again, looking at Goldbach, I should first have wondered whether there are counterexamples in Infinite Integers such as p-adics or AP-adics. There most certainly are counterexamples. A few counterexamples are these: (a) ....191817161514131211109876543210 (b) ....888833338883338833830 (c) .....666666660 And come to think of it, my Galois Algebra proof of Goldbach has restrictions as does all of Projective Geometry has restrictions. So I do learn something new in all of this. The Galois Algebra of Goldbach says that multiplication requires at least two prime factors for Even Naturals >4. If we thus interchange multiplication with addition, every Even Natural requires two prime Goldbach summands. But notice that all of Projective Geometry is finite geometry of objects of lines and points interchangeability. One theorem in projective geometry that talks of points and then interchanges them with lines is that of finite mathematics since points are finite and the lines are line segments. The study of Projective Geometry never made that clear, and only long after having taken the subject might someone, upon reflection, realize that the subject was finite geometry. So that the Galois Algebra proof of Goldbach is not a Goldbach to infinity but a finite Goldbach to 10^500 or less. Goldbach is true for N smaller than 1000 or 10,000, but Goldbach is not true for what "old math called infinity" Goldbach is not true for "out to infinity" unless we specify that 10^500 is the boundary of finite and infinite. This is the same situation for Fermat's Last Theorem and for the Riemann Hypothesis. Now there is going to be alot of carping and complaining by dull people of mathematics, wondering that it appears as if 10^500 is too artificial, since it seems as though Goldbach works just as well out to 10^600 or 10^2,000 and that Fermat's Last Theorem or Riemann Hypothesis are good for that distance also. But the reason they are dullards, is because Physics is exhausted at 10^500 and especially its inverse of 10^-500. Has anyone really, ever noticed that in mathematics, there is a huge bias of theorems for the world at large, but seldom if ever a theorem of the world at the small scale? It is not because of a lack of interest in the microworld for mathematics, but more to do with our inability to think small. And mathematics and mathematicians never really payed attention to math at the small. Occasionally some math is of the microworld such as Poincare Conjecture and the reason PC has never been proven, well, because it too, like Goldbach, like Fermat's Last Theorem and like the Riemann Hypothesis, has run into the boundary between what is finite versus infinite. A long time back in this book which is approaching a thousand pages or more, I gave a proof that you cannot have absolute betweeness in geometry. That you cannot have a axiom that says between any two points on a line there is a new third point. Where does that breakdown? It breaks down at 10^-500. Poincare Conjecture is never provable because the axiom of betweeness stops at 10^-500. Another proof I gave was that if you have absolute betweeness, then you have triangles in Euclidean Geometry whose angle sum is larger than 180 degrees. The point I am making with 10^-500, is that people, dullards in math, look at Goldbach, FLT, RH and think, oh, why stop at 10^500 for those seem to work at 10^600. And the answer to the dullards, is that as you push beyond 10^500, you have to realize that you are pushing beyond 10^-500 in the microworld where you have triangles in that microworld with angle sums larger than 180 degrees. So I have been able to prove Infinitude of Twin Primes, the Polignac conjecture, the Infinitude of Mersenne primes, the Infinitude of Perfect Numbers, and a broad class of infinitude of primes proof, conventionally. I was able to do this because I saw a flaw in the history of mathematics with the inability to properly give a Euclid Indirect method Infinitude of Regular Primes. I also was able to prove that No Odd Perfect Number (except 1) exists by conventional means, where the proof is a template of how square root of 2 is irrational was proven. But as for Goldbach, FLT, and Riemann Hypothesis and the Poincare Conjecture, they are unprovable in their "old math format" because the old-math never defined what it means to be finite-number versus infinite-number. Let me also add, that the Kepler Packing Problem is also unproveable because of the silly ill-defined notion of infinite tiling when packing requires a distinction between finite number and infinite number. So that once KPP is embedded in 10^500, it is instantly proveable that the hexagonal closed packing is not the densest of packings but requires a mix of packings which includes the oblong packing at the container walls. In sum total, mathematics, from Ancient Greece times has built up alot of poor definitions and misconceptions which has led to conjectures that contain those poor-definitions and thus never proveable. And this book is all about cleaning up the messy house of mathematics. 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