Graham's Number

          This weeks post will be on the Graham's number, and what that number means. Grahams number is known to be an immensely large number in the mathematical community, and hopefully this post will help shows its scale, and purpose. First this post will cover what the number refers to, and then it will talk about how absurdly big graham's number really is.

          Graham's Number is a number named after a mathematician by the name of Ron Graham. Graham's number is ridiculously big, and it refers to a geometry problem with respect to higher planes of spatial dimensions. Starting with spatial dimensions, the zero dimensional space refers to a singular point in space. The point has no dimensions, and is a singularity. Moving up in dimensions, the first spatial dimension refers to a connection between two zero dimension singularities via an infinite number of points, or a line segment. The second spatial dimension refers to connecting two lines to created a two dimensional plane. This plane could be pictured as a square, and if all the vertices were connected with lines including the diagonals, there would be six lines. If the lines were colored either red or blue there are six possible configurations. In terms of binomial expansion, there are four vertices, and two colors, so the notation would be four choose two, which are six. That's great, but who cares? The point of the problem is to look for certain avoidable configurations of colors and lines. The conditions that trying to be avoided are four points, that are flat, with six line connections, and all those line connections are the same color. Essentially the pain tis avoid the square that was mentioned earlier being all the same color. Obviously, this is avoidable in two dimensions, because as long as one line is a different color, then the condition is avoidable. However, as the number of spatial dimensions increases, the condition becomes avoidable. The question is which spatial dimensions is the configuration avoidable. Well spatial dimensions 2 through 12 are all avoidable, but all spatial dimensions between 13 and Graham's number of spatial dimensions are unknown. Graham's number of spatial dimensions makes the criteria unavoidable, but if there are smaller numbers of spatial dimensions that are unavoidable is still unknown. 

       So how big is Graham's Number? Graham's number is best explained in a notation called arrow notation. An example of arrow notation would be three arrow three or 3 ^ 3. This means 3 to the third power, or 27. Three double arrow (3^^3) means three arrow, three arrow three, or 3^(3^3), that’s the same as three raised to the twenty-seven, or 7.6 trillion. This is creating a tower of threes, or three rose to the power three raised to the power three. Three triples arrow (3^^^3) means that there is a tower of threes 7.6 billion threes long. The point to take away from this is (3^3) = 27, (3^^3) = 7.6 billion, and (3^^^3) = roughly 1.26 X 10 ^ 3,638,334,640,024. Meaning arrow notation gets big numbers very fast. Three four arrow three (3^^^^3) is a ridiculously large number. Take that crazy number, and that is how many arrow are in the next step, or (3(3^^^^3 number of arrows) 3). Repeat this process again by taking that even bigger number and making it the number of arrows in the next step. Repeat this process 64 times and then you have yourself Graham's number. Unsurprisingly this number once held the world record for the biggest mathematical number used in a proof.

    Graham's number is so large that there are not names for number to describe the number of digits in Graham's Number. Mathematicians understand enough properties of the powers of three to figure the last 500 digits, but it is thought that the first digit will never be known. Now how does a number like this even get figured out? What process lead to this number? As mentioned earlier this number has a meaning, it is the upper bound of number of spatial dimensions that follows a certain criteria. Ron's insight did not allow him to solve the problem, but rather a process that would limit the bound from infinity. This is similar to the post on the twin prime conjecture in that in both situations the method used was not optimized, bath rather the start. However unlike the twin prime conjecture there have not been any recent break through limiting the bounds, but hopefully there will be.

         Sources https://www.youtube.com/watch?v=GuigptwlVHo

                       https://www.youtube.com/watch?v=HX8bihEe3nA

                       http://mathworld.wolfram.com/GrahamsNumber.html

The Twin Prime Conjecture

         This weeks post will be covering a progression of the twin prime gap conjecture. The problem is faulty simple, and is only recently had a major breakthrough. Although there have been breakthroughs the problem remains unsolved. Firstly the problem itself will be covered. Secondly the order of progressions of the problem will be covered in this post.

         The twin prime conjecture is a fairly straightforward conjecture. It has been proven that there are an infinite number of prime numbers. The means that there is an infinite number of numbers that have no divisors other then one and itself. The twin prime conjecture focuses on the gap between prime numbers. There is one exception, but for the most part the smallest gap between two numbers is two. This is because every other number is even, so every other number is divisible by two. The only exception to that rule is, two, the only even prime number. If two prime numbers have a gap of two between them, then they are called twin primes. Primes with a gap of four are called cousin primes, and gaps of six are called sexy primes. The conjecture says that there is an infinite number of town primes. Meaning there is an infinite number of primes with only a gap of two between them. The origins of this problem are unknown. Some individuals say that the work comes from Euclid a couple thousand years ago. This could be very possible, but it was only formally written down a few hundred years ago, so it is at least that old. 

         There have been a number of breakthroughs in recent years towards this problem, but the first big breakthrough, that got the wheels turning happened within the last five years. From the University of New Hampshire a man by the name of Yitang Zhang released a paper putting a limit on the gaps. Before Yitang there was no approach to the problem, what he managed to do was say that there is an invite number of primes with gap N, N is between one and seventy million. To be clear he did not prove that the gap between primes are seventy million or less, but rather that there is an infinite number of primes with gap of seventy million. This breakthrough allowed for the first time a method of whittling down this seventy million to two. Yitang's result was the breakthrough, but it was not optimized. This means that though his result was seventy million, it was also understood that by tweaking the argument the bounds could shrink. This lead to a competition to see who could tweak the argument in such a way, to get the smallest bound. These developments lead to the gap being shrunk to 4,680, which is a big improvement, but not quite the end goal. Though the twin prime conjecture has not been proven, there have been more recent results that whittle down the gap number even more. Before the work of Yitang, there was an approach to solve the problem by mathematicians Goldston, Pintz, and Yildirim. Their work was important, but it required outside proofs, meaning that this works, but only if this other problem is solved. By sorting through these hiccups of the work of Goldstone, pints, and Yildirim, James Mayard, and Terrence Tao were able to whittle it down even further. Their method was completely different to Yitang, and both developed it simultaneously, and separately. Thus their work was named the Mayard-Tao method, and it brought the gap down to 256. 

         Even though this newly found method offers new insight into the problem, it has a flaw. The method used has a limit to it. If the method is completely optimized, it sill can not be used to prove the twin prime conjecture. Theoretically the Mayard-Tao method can only bring the gap down to there being an infinite number of primes with a gap of six. So even if the this method is completely optimized it is only a victory for the sexy primes. The win prime conjecture still needs a new approach, or new idea, to be solved.




Sources https://www.youtube.com/watch?v=vkMXdShDdtY
                  https://www.youtube.com/watch?v=QKHKD8bRAro
                  http://mathworld.wolfram.com/TwinPrimeConjecture.html