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Not long ago I finded Goodstein Functions pretty amazing, and it lead me to make this function. The Function is expressed as B(j)=k, where j is the seed, which is the number we're dealing with, and k is the length of the sequence â€“ the # of steps it takes to reach 0. It is hard explain, but I shall my best.
I use square brackets around the number in any base b, and space between digit to avoid confusion when the base is higher than 10. It means for instance that [1 2]_{13} â‰ [12]_{13}, since the first one is equal to 15 and the second one is equal 12, and therefor not the same.
The seed number is what we are starting, and it need to be written in base 2. Example: if the seed is 3 then you write it as 11 in base 2 or in the Batrix way of writing it:â€¦
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The 1, 2, 3, 4 game
This game is similar to the four fours game, but we're using numbers from 1 to 4 instead. The rules are almost the same as four fours game, but this includes also fgh, double knuth arrow, beaf, hyperE aswell, and exponetial factorials. if using fgh, you must use the given numbers. The most important rule is that numbers 1 to 4 must be in ascending order.
Allowed mathematical notation: Normal mathematic (+, , x, /, (), !) More factorials (x!!, X!!!,...) Subfactorial (!x) Roots and Exponents (âˆš, ^) (nth roots is not allowed, only squareroots) Î“(x) beaf {x, y, z} or {a, b, c, d} hyperE (Ex or Ex#y ...) â†‘ or more arrows Æ’gh (this is optional. you can have 1 as subscript and 2 as superscript, or you may have 2 as sub. and â€¦
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Let me introduce the function based on factorial (!). I will call this function Super Staging Factorial Function. In this function we can go far beyond with the simple notation !. I gonna tell how I got this idea. I mainly got this inspiration by dysfunctionalfactorial's idea at http://www.artofproblemsolving.com/community/c1664h1003843 about factorial. This is not be confused with Faxul terms. The naming system focuses with the number 200, similar to Faxul terms. I use Faxul terms to see how fast this function grow perspective. I don't think this function would beat Expofaxul terms very fast. Numbers I have created is based on the name Lexol, meaning "Level Factorial", the first tier. And name Raxol, meaning "Rank Factorial", the second tiâ€¦
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