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• The HyperRex Function

April 29, 2018 by B1mb0w

My new HyperRex function is compared here to recursive functions such as Veblen and my previous Hyper function. The HyperRex function is a set of two functions $$H()$$ and $$r()$$ and has a growth rate well beyond $$f_{\vartheta(\Omega\uparrow\uparrow\omega)}(n)$$

The notation I use here is not in general use, but I find helpful. They are parameter subscript brackets, leading zeros assumption, recursion parameter subscript $$*$$, and the decremented function $$C$$.

Parameter Subscript Brackets, where:

$$M(a,0_{[2]}) = M(a,0,0)$$

$$M(a,0_{[2]},b_{[3]},1) = M(a,0,0,b_1,b_2,b_3,1)$$

$$M(0_{[x]},0_{[2]},b_{[3]},1) = = M(0_{[x + 2]},b_1,b_2,b_3,1) = M(b_1,b_2,b_3,1)$$

Recursion Parameter Subscript $$*$$, where:

$$M^2(a)… Read more > • The Hyper Function April 28, 2018 by B1mb0w This blog will compare recursive functions such as Veblen to my new Hyper function. The Hyper function is a set of two functions \(H()$$ and $$p()$$ and has a growth rate $$\approx f_{\vartheta(\Omega\uparrow\uparrow\omega)}(n)$$

The notation I use here is not in general use, but I find helpful. They are parameter subscript brackets, leading zeros assumption, recursion parameter subscript $$*$$, and the decremented function $$C$$.

Parameter Subscript Brackets, where:

$$M(a,0_{[2]}) = M(a,0,0)$$

$$M(a,0_{[2]},b_{[3]},1) = M(a,0,0,b_1,b_2,b_3,1)$$

$$M(0_{[x]},0_{[2]},b_{[3]},1) = M(b_1,b_2,b_3,1)$$

Recursion Parameter Subscript $$*$$, where:

$$M^2(a) = M^2(a_*) = M(M(a))$$ and $$M(a,b_*) = M(a,b)$$

$$M^2(a,b_*) = M(a,… Read more > • Comparison of Recursive Functions April 21, 2018 by B1mb0w This blog will compare recursive functions such as Veblen and my own Big number and T-Rex functions. The notation I use here is not in general use, but I find helpful. They are parameter subscript brackets, leading zeros assumption, recursion parameter subscript \(*$$, and the decremented function $$C$$.

Parameter Subscript Brackets, where:

$$M(a,0_{[2]}) = M(a,0,0)$$

$$M(a,0_{[2]},b_{[3]},1) = M(a,0,0,b_1,b_2,b_3,1)$$

$$M(0_{[x]},0_{[2]},b_{[3]},1) = M(b_1,b_2,b_3,1)$$

Recursion Parameter Subscript $$*$$, where:

$$M^2(a) = M^2(a_*) = M(M(a))$$ and $$M(a,b_*) = M(a,b)$$

$$M^2(a,b_*) = M(a,M(a,b))$$

$$M^2(a_*,b) = M(M(a,b),b)$$

Decremented Function $$C$$, where for any function:

$$M(a_{[b]},c + 1,d_{[e]})$$ then $$C = M… Read more > • The Big number function April 10, 2018 by B1mb0w The Big number function is a very fast growing function. It's growth rate is well beyond \(f_{LVO}(n)$$.

The Big number function is a pair of functions $$B()$$ and $$g()$$ which use this simple rule set:

$$B(n) = B(0,n) = n + 1$$

$$B(a + 1, n) = B^n(a,n_*)$$

$$B(g(0), n) = B(n,n)$$ and other instances of $$n$$ can be substituted with $$g(0)$$

$$g(c + 1) = g(0, c + 1) = B^{g(c)}(g(c)_*,g(c))$$

and

$$g(1, 0_{[d + 1]}) = g^{g(1, 0_{[d]})}(1_*, 0_{[d]})$$

$$g(b + 1, 0) = g^{g(b,0)}(b,0_*)$$

$$g(b, c + 1) = B^{g(b,c)}(g(b,c)_*,g(b,c))$$

and

$$g() = g_0()$$

$$g_{a + 1}(0) = g_a(1, 0_{[g_a(0)]})$$

$$g_a(c + 1) = B^{g_a(c)}(g_a(c)_*,g_a(c))$$

$$g_a(b + 1, 0) = g_a^{g_a(b,0)}(b,0_*)$$

$$g_a(b, c + 1) = B^{g_a(b,c)}(g_a(b,c)_*,g_a(b,c))$$

$$g_a(1, 0, 0) = g_a^{g_a(… Read more > • The Jurassic Number April 9, 2018 by B1mb0w The Jurassic Number is one of the largest named numbers. I calculate it is \(\approx f_{\vartheta(\Omega^{\omega^2.2})}(65500000)$$.

It is calculated using the Jurassic Function $$Jurassic(n)$$.

The Jurassic Function is defined using The T-Rex Function and this simple definition:

$$Jurassic(n) = T(1,0_{[n]},n)$$ using parameter subscript brackets $$[n]$$.

Parameter subscript brackets are useful for functions with many parameters, such as:

$$M(a,0_{[1]}) = M(a,0)$$

$$M(a,0_{[3]}) = M(a,0,0,0)$$

$$M(a,b_{[2]}) = M(a,b_1,b_2)$$

$$M(a,0_{[2]},b_{[3]},1) = M(a,0,0,b_1,b_2,b_3,1)$$

I calculate this T-Rex function to equal:

$$T(1,0_{[n]},n) \approx f_{\vartheta(\Omega^{\omega^2.2})}(n)$$

The Jurassic Number is equal to a T-Rex Function with 65,500,000 (65.5…