On Thursday, April 14, 2016 at 2:23:39 PM UTC-5, Archimedes Plutonium wrote: > On Wednesday, April 13, 2016 at 11:41:03 PM UTC-5, Archimedes Plutonium wrote: > (snipped) > > > > Alright this gives me an idea to pursue for the "perfect Legendre prime that is nestled in between two successive perfect squares. Say I wanted the Legendre prime for 9 to 16, then I go to 16 itself and the 16 Array and find the Goldbach Multiplication GM primes to be 13 and 3 so that 13 is the Legendre prime. Now I look at 16 to 25 for the Legendre prime and go to the 26 Even Number Array and the GM primes are 23 with 3 where 23 is the Legendre prime. Now I want the Legendre prime for 25 to 36 and go into the 36 Even Number Array and find the GM primes to be 29 with 7 so that 29 is the perfect Legendre prime. > > > > In this we see that we need prove GA, Goldbach Addition first and as a corollary we prove Goldbach Multiplication GM and as a further corollary prove the Legendre conjecture. > > > > Now the proof of GA was really quite simple, but is the proof of GM more difficult? Not really because in proving GA we accepted primes such as 3+3 and 5+5 and 7+7 and to prove GM we cannot accept those primes as a Goldbach solution but the pair of primes has to be different, so that we cannot accept 5+5 for the 10 Array but can accept 3+7 or 3*7 as the GM solution where 7 becomes, also the Legendre solution for perfect squares 4 to 9. > > > > Alright, after last night doing the method for finding the perfect Legendre prime, I realized something else that is very important and reveals the hidden structure of prime numbers. > > It is fair to say that Even Number Arrays is the greatest advancement of Number theory in mathematics since Galois of the 1830s. That Number theory is in a massive advancement simply from the discovery of Even Number Arrays which pits addition alongside multiplication. > > I say massive advancement for the Arrays reveal the structure of primes compared to other numbers. The primes do have a pattern which this 3,5,7 Staircase reveals. If I had to put this pattern into words, I would say the pattern is of "ultra economic savings". For example, there are just enough primes to yield all the Even Numbers by a Goldbach Multiplication GM. Remember GM requires two different primes, some are Legendre primes in the Array. So that the Legendre primes for the perfect squares 81 to 100 would be 3*97 with 97 the Legendre prime. > > Now, what I mean by economic savings is that there exists just so many primes as to have all the even numbers satisfied by a GM pair for all even numbers starting with 6. > > STAIRCASE NUMBER THEORY with EVEN NUMBER ARRAYS: Let me show you how this works and why 9 cannot be a prime number. So we start with the number 6 and what we have are only 3,5,7 to work with so we ask can we get to 6 with just 3,5,7? And the answer is yes in that of 3+3. Now can we get to all the even numbers from 6 to 100 using only 3,5,7? And the answer is yes, in that 6 = 3+3, 8 is 3+5, 10 is 5+5, 12 is 5+7, . . , 50 is 3+47, 52 is 5+47, 54 is 7+47, 56 is 3+53, . . 100 is 3+97, onwards until we reach 113 and 127 which is the largest gap in primes so far. A gap spacing of 14 units. So obviously my staircase of 3,5,7 needs the addition of new primes of capture 122 to 130 even numbers and here I include into the staircase the primes 3,5,7,11,13,17,19 to capture every even number from 6 to at least 1000. Actually, I do not know at what even number these seven primes require me to include 23 to the staircase. > > What the staircase method tells us is that 9 can not be a prime number, even though we knew that before, but cannot be prime for 9 is never needed to build a even number, when 3,5,7 do all the work that is needed. > > So, in other words, primes are never excessive or redundant in that we do not need 9 as prime. > > So, what is the Staircase Conjecture of mine? > > Simply stated, give me a Even Number at random, and depending on the size of that even number, if between 0 and 100, the Legendre prime is composed of two primes where the smallest is one of 3,5,7. If between 100 and 1000 the Legendre prime is composed of two primes where the smallest is one of 3,5,7,11,13,17,19. For in the case of 128, I need 19 + 109 with 109 as the Legendre prime. > > Now, I see something very tantalizing in that the staircase primes seem to cluster around 10 and then 100. Do they cluster at 1000, just a little ways beyond 1000? If they do, I speculate it is because of pi. In that 10 and 100 are multiplies of sqrt10 which is close to pi in that sqrt10 is 3.16... which is close to pi of 3.14... > > SUMMARY: The primes can build all the even numbers starting with 6 by a staircase of addition, starting with just 3,5,7 and then at 122 adding primes 11,13,17,19 to the staircase and then at some further larger even number adding more primes to the staircase primes. > > > 4Research > 4Disclaimer: Due to the unconventional and speculative nature of this posting and thread, it would be inadvisable for students to apply any of the contents to their school course work. > > 4AP
Okay, here's my randomly selected even number: 10^100 + 100368.