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  • Nayuta Ito

    Intuitive Arrow Notation

    October 17, 2017 by Nayuta Ito

    This notation is supposed to express Linear Omega Level intuitively, so don't expect it to go any higher.


    a[c]b is supposed to be is an array.

    a[#,1]b=a[#]b

    a[#,n,m]b=a[#,a[#,n]b,m-1]b


    3->3->64->2=3[27,63]3=3[1,64]4

    Graham's number=3[4,64]3

    3->3->65->2=3[27,64]3

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  • Nayuta Ito

    zootzoot prefix

    October 3, 2017 by Nayuta Ito

    As you know, zootzoot is googol!1, zootzootplex is googolplex!1, and zootzootduplex is googolduplex!1. Then, we can see the rule:

    Rule: Zootzootα is googolα!1 using hyperfactorial array notation, where α is any string while googolα is well-defined.

    The zootzootzootplex looks like the exception, but since googolzootplex is not defined, the rule is not applied to this.

    Here are some numbers:

    • zootzooti=(102^51)!1, derived from googoli.
    • zootzootij=(104^52)!1, derived from googolij.
    • zootzootiji=(106^53)!1, derived from googoliji.
    • zootzootiv=(108^54)!1, derived from googoliv.
    • zootzootv=(110^55)!1, derived from googolv.
    • zootzootex=(120^60)!1, derived from googolex.
    • zootzootbang=((10^100)!)!1, derived from googolbang.
    • zootzootding=(10^500)!1, derived from go…
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  • Nayuta Ito

    Many googol series articles have the link to this page, but actually, it's outdated. So, I'm going to summarize the new "improved" page.


    The definition of Ackermann's Generalized Exponential Notation, which is used throughout the page, says:

    g(0, a, b) = b + a (addition) g(1, a, b) = ab = a*b = a++b (multiplication) g(2, a, b) = ba = a^b = a↑b = a**b = a+++b (exponentiation) g(3, a, b} = ab = a^^b = a↑↑b (tetration) g(a, 0, 1) = 1 (zeroth power) g(c, 0, a) = 1 (zeroth operation) g(a, b, c) = g(a - 1, g(a -1 , b, c) , c) (expansion) g(a, b, c, d) = g(a - 1, b, c, g(b, c, d)) (nesting about base number) g(a, b, c, d, e) = g(a - 1, b, g(c, d, e), e) (nesting about power number) g(a, b, c, d., e, f) = g(a - 1, b, c, g(d, e, f), e, f) (nesting abou…
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  • Nayuta Ito

    SIR(1,n,d) is exactly d and you cannot go higher. Here's the proof:

    There is a theorem that if a polynomial has only integer coefficients, all the coefficients of the factors of it are integers.

    If an equation whose root goes higher than d (let it e), the absolute value of the of the constant term of the original equation must be e or greater.

    The exception to the statement above is when one of the roots is zero but in that case, you can divide the whole equation by x and start it over.

    Anyway, the absolute value of the constant term (or the lowest non-zero term) must be e or greater, which means it is bigger than d.

    It's a contradiction because we can only use the integer between -d and d for the original coefficients.

    Therefore, the absolute o…

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  • Nayuta Ito



    0 1 2 3 4 5 6 7 8 9
    2.001687
    2.003536
    2.005359
    2.007157
    2.008930
    2.010680
    2.012406
    2.014109
    2.015790
    2.017450
    2.019088
    2.020705
    2.022302
    2.023879
    2.025437
    2.026976
    2.028496
    2.029998
    2.031482
    2.032949
    2.034398
    2.035831
    2.037248
    2.038648
    2.040033
    2.041403
    2.042757
    2.044097
    2.045422
    2.046733
    2.048030
    2.049313
    2.050583
    2.051839
    2.053083
    2.054314
    2.055533
    2.056739
    2.057933
    2.059116
    2.060286
    2.061446
    2.062594
    2.063731
    2.064858
    2.065974
    2.067079
    2.068174
    2.069259
    2.070334
    2.071399
    2.072455
    2.073501
    2.074538
    2.075566
    2.076584
    2.077594
    2.078595
    2.079588
    2.080572
    2.081547
    2.082515
    2.083474
    2.084426
    2.085369
    2.086305
    2.087233
    2.088154
    2.089067
    2.089973
    2.090872
    2.091764
    2.092649
    2.093527
    2.094398
    2.095262
    2.096120
    2.096971
    2.097816
    2.098655
    2.099487
    2.100313
    2.101133
    2.101947
    2.102755
    2.103557
    2.104354
    2.105144
    2.105930
    2.106709
    2.107483
    2.…






























































































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