Latest revision as of 15:08, 18 August 2025

Machine: 0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD (bbch)

Better lower bound for BB(4,3)

Definitions

$f(n)=2^{2^{n+1}}$

$g(n)={\frac {5\times 2^{2^{f^{n}(0)+1}+2}-8}{9}}$

$n0={\frac {5\times 2^{2^{2^{32}+1}+1}-4}{9}}$

$n1=2^{2^{32}+1}-4$

Σ = $5\times 2^{2^{f^{g^{n1}(n0)}(0)+1}+2}+7$

Lower bound on f^n(0)

$f(n)=2^{2^{n+1}}$

$f(0)=2^{2^{1}};f(f(0))=2^{2^{(2^{2^{1}}+1)}}>(2\uparrow )^{4}1;f^{3}(0)=2^{2^{(2^{2^{(2^{2^{1}}+1)}}+1)}}>(2\uparrow )^{6}1$

$f^{k}(0)=2^{2^{f^{k-1}(0)+1}}>2^{2^{f^{k-1}(0)}}=>(2\uparrow )^{2}f^{k-1}(0)=>(2\uparrow )^{2a}f^{k-a}(0)$

$=>f^{k}(0)>(2\uparrow )^{2k}1=2\uparrow \uparrow 2k$

$=>f^{n}(0)>2\uparrow \uparrow 2n$

Upper bound

$f_{u}(n)=2^{2^{2^{n}}};f_{u}(0)=2^{2^{2}};f_{u}^{0}(0)=1$

$f_{u}(n)>f(n)$ as $2^{n}>n+1$

$f_{u}^{k}(0)=(2\uparrow )^{3}f_{u}^{k-1}(0)$

$f_{u}^{2}(0)=f_{u}(2^{2^{2}})=2^{2^{2^{2^{2^{2}}}}}=>(2\uparrow )^{6}1$

$f_{u}^{k}(0)=(2\uparrow )^{3k}1=2\uparrow \uparrow 3k;f_{u}(n)>f(n)=>2\uparrow \uparrow 3k>f(n)$

$f^{n}(0)<2\uparrow \uparrow 3n$

Lower bound on g^k(n)

$g(n)={\frac {5\times 2^{2^{f^{n}(0)+1}+2}-8}{9}}>{\frac {5\times 2^{2^{2\uparrow \uparrow 2n}+2}-8}{9}}>2^{2^{2\uparrow \uparrow 2n}}=2\uparrow \uparrow (2n+2)$

$g^{2}(n)>{\frac {5\times 2^{2^{f^{2\uparrow \uparrow (2n+2)}(0)+1}+2}-8}{9}}>{\frac {5\times 2^{2^{2\uparrow \uparrow 2\uparrow \uparrow (2n+2)}+2}-8}{9}}>2\uparrow \uparrow 2\uparrow \uparrow (2n+2)$

$g^{k}(n)>2\uparrow \uparrow (2\times g^{k-1}(n)+2)=(2\uparrow \uparrow )^{1}(2\times g^{k-1}(n)+2)=>>(2\uparrow \uparrow )^{a}(2\times g^{k-a}(n)+2)$

$g^{k}(n)>(2\uparrow \uparrow )^{k}(2n+2)>(2\uparrow \uparrow )^{k}2n$

Upper bound

$g(n)={\frac {5\times 2^{2^{f^{n}(0)+1}+2}-8}{9}}<{\frac {5\times 2^{2^{(2\uparrow \uparrow 3n)+1}+2}-8}{9}}<5\times 2^{2^{(2\uparrow \uparrow 3n)+1}+2}<2\uparrow \uparrow (3n+3)$

$g^{2}(n)<2\uparrow \uparrow (3\times (2\uparrow \uparrow (3n+3))+3)<2\uparrow \uparrow 2\uparrow \uparrow (3n+4)$

$g^{k}(n)<2\uparrow \uparrow (3\times (g^{k-1}(n))+3)<2\uparrow \uparrow 2\uparrow g^{k-1}(n)$

$g^{k}(n)<(2\uparrow \uparrow )^{k}(3n+k+2)$

Lower bound on g^n1(n0)

$g^{n1}(n0)=g^{2^{2^{32}+1}-4}({\frac {5\times 2^{2^{2^{32}+1}+1}-4}{9}})>g^{2^{2^{32}+1}-4}(2^{2^{2^{32}}})>(2\uparrow \uparrow )^{2^{2^{32}+1}-4}2^{2^{2^{32}}}$ ; Note that $2\uparrow \uparrow 6<2^{2^{2^{32}}}<2\uparrow \uparrow 7$

$(2\uparrow \uparrow )^{2^{2^{32}+1}-4}2^{2^{2^{32}}}>(2\uparrow \uparrow )^{2^{2^{32}+1}-4}2\uparrow \uparrow 6=>(2\uparrow \uparrow )^{2^{2^{32}+1}-3}6>(2\uparrow \uparrow )^{2^{2^{32}+1}-3}4=>(2\uparrow \uparrow )^{2^{2^{32}+1}-3}2\uparrow \uparrow 2=>(2\uparrow \uparrow )^{2^{2^{32}+1}-2}2$

$g^{n1}(n0)>(2\uparrow \uparrow )^{2^{2^{32}+1}-2}2$

Upper bound

$g^{n1}(n0)=g^{2^{2^{32}+1}-4}({\frac {5\times 2^{2^{2^{32}+1}+1}-4}{9}})<(2\uparrow \uparrow )^{2^{2^{32}+1}-4}(3\times ({\frac {5\times 2^{2^{2^{32}+1}+1}-4}{9}})+2^{2^{32}+1}-4+2)<(2\uparrow \uparrow )^{2^{2^{32}+1}-4}2\uparrow \uparrow 7=>(2\uparrow \uparrow )^{2^{2^{32}+1}-3}7$

$g^{n1}(n0)<(2\uparrow \uparrow )^{2^{2^{32}+1}-3}7$

or, for a more precise upper bound:

$g^{n1}(n0)=g^{2^{2^{32}+1}-4}({\frac {5\times 2^{2^{2^{32}+1}+1}-4}{9}})<(2\uparrow \uparrow )^{2^{2^{32}+1}-4}(3\times ({\frac {5\times 2^{2^{2^{32}+1}+1}-4}{9}})+2^{2^{32}+1}-4+2)<(2\uparrow \uparrow )^{2^{2^{32}+1}-4}(2^{2^{2^{33}}})$

$g^{n1}(n0)<(2\uparrow \uparrow )^{2^{2^{32}+1}-4}(2^{2^{2^{33}}})$

or, for an even more precise upper bound:

$(2\uparrow \uparrow )^{2^{2^{32}+1}-4}(3\times ({\frac {5\times 2^{2^{2^{32}+1}+1}-4}{9}})+2^{2^{32}+1}-4+2)<(2\uparrow \uparrow )^{2^{2^{32}+1}-4}(2^{2^{2^{32}+2}})$

$g^{n1}(n0)<(2\uparrow \uparrow )^{2^{2^{32}+1}-4}(2^{2^{2^{32}+2}})$

Lower bound on f^g^n1(n0)(0)

$g^{n1}(n0)>(2\uparrow \uparrow )^{2^{2^{32}+1}-2}2$ and $f^{n}(0)>2\uparrow \uparrow 2n$

$f^{g^{n1}(n0)}(0)>f^{(2\uparrow \uparrow )^{2^{2^{32}+1}-2}2}(0)>2\uparrow \uparrow (2\times (2\uparrow \uparrow )^{2^{2^{32}+1}-2}2)>2\uparrow \uparrow (2\uparrow \uparrow )^{2^{2^{32}+1}-2}2=>(2\uparrow \uparrow )^{2^{2^{32}+1}-1}2$

$(2\uparrow \uparrow )^{2^{2^{32}+1}-1}2=>(2\uparrow \uparrow )^{2^{2^{32}+1}-1}2\uparrow \uparrow 1=>(2\uparrow \uparrow )^{2^{2^{32}+1}}1=>2\uparrow \uparrow \uparrow 2^{2^{32}+1}$

$f^{g^{n1}(n0)}(0)>2\uparrow \uparrow \uparrow 2^{2^{32}+1}$

or, for a more precise lower bound:

$g^{n1}(n0)>(2\uparrow \uparrow )^{2^{2^{32}+1}-4}2^{2^{2^{32}}}$

$f^{g^{n1}(n0)}(0)>f^{(2\uparrow \uparrow )^{2^{2^{32}+1}-4}2^{2^{2^{32}}}}(0)>2\uparrow \uparrow (2\times (2\uparrow \uparrow )^{2^{2^{32}+1}-4}2^{2^{2^{32}}})>2\uparrow \uparrow (2\uparrow \uparrow )^{2^{2^{32}+1}-4}2^{2^{2^{32}}}=>(2\uparrow \uparrow )^{2^{2^{32}+1}-3}2^{2^{2^{32}}}$

$f^{g^{n1}(n0)}(0)>(2\uparrow \uparrow )^{2^{2^{32}+1}-3}2^{2^{2^{32}}}$

Upper bound

$g^{n1}(n0)<(2\uparrow \uparrow )^{2^{2^{32}+1}-3}7$ and $f^{n}(0)<2\uparrow \uparrow 3n$

$f^{g^{n1}(n0)}(0)<2\uparrow \uparrow (3\times (2\uparrow \uparrow )^{2^{2^{32}+1}-3}7)<2\uparrow \uparrow (2\uparrow \uparrow )^{2^{2^{32}+1}-3}8=>(2\uparrow \uparrow )^{2^{2^{32}+1}-2}8$

$g^{n1}(n0)<(2\uparrow \uparrow )^{2^{2^{32}+1}-2}8$

or, for a more precise upper bound:

$g^{n1}(n0)<(2\uparrow \uparrow )^{2^{2^{32}+1}-4}(2^{2^{2^{33}}})$ and $f^{n}(0)<2\uparrow \uparrow 3n$

$f^{g^{n1}(n0)}(0)<2\uparrow \uparrow (3\times (2\uparrow \uparrow )^{2^{2^{32}+1}-4}(2^{2^{2^{33}}}))<2\uparrow \uparrow (2\uparrow \uparrow )^{2^{2^{32}+1}-4}(2^{2^{2^{33}}}+1)=>(2\uparrow \uparrow )^{2^{2^{32}+1}-3}(2^{2^{2^{33}}}+1)$

$f^{g^{n1}(n0)}(0)<(2\uparrow \uparrow )^{2^{2^{32}+1}-3}(2^{2^{2^{33}}}+1)$

or, for an even more precise upper bound:

$g^{n1}(n0)<(2\uparrow \uparrow )^{2^{2^{32}+1}-4}(2^{2^{2^{32}+2}})$ and $f^{n}(0)<2\uparrow \uparrow 3n$

$f^{g^{n1}(n0)}(0)<2\uparrow \uparrow (3\times (2\uparrow \uparrow )^{2^{2^{32}+1}-4}(2^{2^{2^{32}+2}}))<2\uparrow \uparrow (2\uparrow \uparrow )^{2^{2^{32}+1}-4}(2^{2^{2^{32}+2}}+1)=>(2\uparrow \uparrow )^{2^{2^{32}+1}-3}(2^{2^{2^{32}+2}}+1)$

$f^{g^{n1}(n0)}(0)<(2\uparrow \uparrow )^{2^{2^{32}+1}-3}(2^{2^{2^{32}+2}}+1)$

Lower bound on Σ

Σ = $5\times 2^{2^{f^{g^{n1}(n0)}(0)+1}+2}+7>5\times 2^{2^{(2\uparrow \uparrow \uparrow 2^{2^{32}+1})+1}+2}+7>2\uparrow \uparrow \uparrow 2^{2^{32}+1}$

Σ > $2\uparrow \uparrow \uparrow 2^{2^{32}+1}$

or, for a more precise lower bound:

Σ = $5\times 2^{2^{f^{g^{n1}(n0)}(0)+1}+2}+7>2^{2^{(2\uparrow \uparrow )^{2^{2^{32}+1}-3}2^{2^{2^{32}}}}}>(2\uparrow \uparrow )^{2^{2^{32}+1}-3}2^{2^{2^{32}}}$

Σ > $(2\uparrow \uparrow )^{2^{2^{32}+1}-3}2^{2^{2^{32}}}$

Upper bound

Σ = $5\times 2^{2^{f^{g^{n1}(n0)}(0)+1}+2}+7<5\times 2^{2^{((2\uparrow \uparrow )^{2^{2^{32}+1}-2}8)+1}+2}+7<2^{2^{2^{(2\uparrow \uparrow )^{2^{2^{32}+1}-2}8}}}<(2\uparrow \uparrow )^{2^{2^{32}+1}-2}9<(2\uparrow \uparrow )^{2^{2^{32}+1}-2}65536=>(2\uparrow \uparrow )^{2^{2^{32}+1}-2}2\uparrow \uparrow 4=>(2\uparrow \uparrow )^{2^{2^{32}+1}-1}4$

$(2\uparrow \uparrow )^{2^{2^{32}+1}-1}4=>(2\uparrow \uparrow )^{2^{2^{32}+1}-1}2\uparrow \uparrow 2=>(2\uparrow \uparrow )^{2^{2^{32}+1}}2=>(2\uparrow \uparrow )^{2^{2^{32}+1}}2\uparrow \uparrow 1=>(2\uparrow \uparrow )^{2^{2^{32}+1}+1}1=>2\uparrow \uparrow \uparrow (2^{2^{32}+1}+1)$

Σ < $2\uparrow \uparrow \uparrow (2^{2^{32}+1}+1)$

or, for a more precise upper bound:

Σ = $5\times 2^{2^{f^{g^{n1}(n0)}(0)+1}+2}+7<2^{2^{2^{(2\uparrow \uparrow )^{2^{2^{32}+1}-3}(2^{2^{2^{33}}}+1)}}}<(2\uparrow \uparrow )^{2^{2^{32}+1}-3}(2^{2^{2^{33}}}+2)$

Σ < $(2\uparrow \uparrow )^{2^{2^{32}+1}-3}(2^{2^{2^{33}}}+2)$

or, for an even more precise upper bound:

Σ = $5\times 2^{2^{f^{g^{n1}(n0)}(0)+1}+2}+7<2^{2^{2^{(2\uparrow \uparrow )^{2^{2^{32}+1}-3}(2^{2^{2^{32}+2}}+1)}}}<(2\uparrow \uparrow )^{2^{2^{32}+1}-3}(2^{2^{2^{32}+2}}+2)$

Σ < $(2\uparrow \uparrow )^{2^{2^{32}+1}-3}(2^{2^{2^{32}+2}}+2)$

General bound on Σ

$2\uparrow \uparrow \uparrow 2^{2^{32}+1}$ < Σ < $2\uparrow \uparrow \uparrow (2^{2^{32}+1}+1)$

More precisely: $(2\uparrow \uparrow )^{2^{2^{32}+1}-2}6$ < Σ < $(2\uparrow \uparrow )^{2^{2^{32}+1}-2}7$

Even more precisely: $(2\uparrow \uparrow )^{2^{2^{32}+1}-2}6<(2\uparrow \uparrow )^{2^{2^{32}+1}-3}2^{2^{2^{32}}}$ < Σ < $(2\uparrow \uparrow )^{2^{2^{32}+1}-3}(2^{2^{2^{32}+2}}+2)<(2\uparrow \uparrow )^{2^{2^{32}+1}-3}(2^{2^{2^{33}}}+2)<(2\uparrow \uparrow )^{2^{2^{32}+1}-2}7$

@@ Line 62: / Line 62: @@
 '''Upper bound'''
-<math>g^{n1}(n0) = g^{2^{2^{32}+1}-4}(\frac {5 \times 2^{2^{2^{32}+1}+1}-4}{9}) < (2 \uparrow\uparrow)^{2^{2^{32}+1}-4}(3 \times (\frac {5 \times 2^{2^{2^{32}+1}+1}-4}{9}) + 2^{2^{32}+1}-4 + 2 < (2 \uparrow\uparrow)^{2^{2^{32}+1}-4}2 \uparrow\uparrow 7 => (2 \uparrow\uparrow)^{2^{2^{32}+1}-3}7</math>
+<math>g^{n1}(n0) = g^{2^{2^{32}+1}-4}(\frac {5 \times 2^{2^{2^{32}+1}+1}-4}{9}) < (2 \uparrow\uparrow)^{2^{2^{32}+1}-4}(3 \times (\frac {5 \times 2^{2^{2^{32}+1}+1}-4}{9}) + 2^{2^{32}+1}-4 + 2) < (2 \uparrow\uparrow)^{2^{2^{32}+1}-4}2 \uparrow\uparrow 7 => (2 \uparrow\uparrow)^{2^{2^{32}+1}-3}7</math>
 <math>g^{n1}(n0) < (2 \uparrow\uparrow)^{2^{2^{32}+1}-3}7</math>
+or, for a more precise upper bound:
+<math>g^{n1}(n0) = g^{2^{2^{32}+1}-4}(\frac {5 \times 2^{2^{2^{32}+1}+1}-4}{9}) < (2 \uparrow\uparrow)^{2^{2^{32}+1}-4}(3 \times (\frac {5 \times 2^{2^{2^{32}+1}+1}-4}{9}) + 2^{2^{32}+1}-4 + 2) < (2 \uparrow\uparrow)^{2^{2^{32}+1}-4}(2^{2^{2^{33}}})</math>
+<math>g^{n1}(n0) < (2 \uparrow\uparrow)^{2^{2^{32}+1}-4}(2^{2^{2^{33}}})</math>
+or, for an even more precise upper bound:
+<math>(2 \uparrow\uparrow)^{2^{2^{32}+1}-4}(3 \times (\frac {5 \times 2^{2^{2^{32}+1}+1}-4}{9}) + 2^{2^{32}+1}-4 + 2) < (2 \uparrow\uparrow)^{2^{2^{32}+1}-4}(2^{2^{2^{32}+2}})</math>
+<math>g^{n1}(n0) < (2 \uparrow\uparrow)^{2^{2^{32}+1}-4}(2^{2^{2^{32}+2}})</math>
 ==Lower bound on f^g^n1(n0)(0)==
 <math>g^{n1}(n0) > (2 \uparrow\uparrow)^{2^{2^{32}+1}-2}2</math> and <math>f^{n}(0) > 2 \uparrow\uparrow 2n</math>
@@ Line 76: / Line 88: @@
 or, for a more precise lower bound:
-<math>g^{n1}(n0) > (2 \uparrow\uparrow)^{2^{2^{32}+1}-3}6</math>
+<math>g^{n1}(n0) > (2 \uparrow\uparrow)^{2^{2^{32}+1}-4}2^{2^{2^{32}}}</math>
-<math>f^{g^{n1}(n0)}(0) > f^{(2 \uparrow\uparrow)^{2^{2^{32}+1}-3}6}(0) > 2 \uparrow\uparrow (2 \times (2 \uparrow\uparrow)^{2^{2^{32}+1}-3}6) > 2\uparrow\uparrow (2 \uparrow\uparrow)^{2^{2^{32}+1}-3}6 => (2  \uparrow\uparrow)^{2^{2^{32}+1}-2}6</math>
+<math>f^{g^{n1}(n0)}(0) > f^{(2 \uparrow\uparrow)^{2^{2^{32}+1}-4}2^{2^{2^{32}}}}(0) > 2 \uparrow\uparrow (2 \times (2 \uparrow\uparrow)^{2^{2^{32}+1}-4}2^{2^{2^{32}}}) > 2\uparrow\uparrow (2 \uparrow\uparrow)^{2^{2^{32}+1}-4}2^{2^{2^{32}}} => (2  \uparrow\uparrow)^{2^{2^{32}+1}-3}2^{2^{2^{32}}}</math>
-<math>f^{g^{n1}(n0)}(0) > (2 \uparrow\uparrow)^{2^{2^{32}+1}-2}6</math>
+<math>f^{g^{n1}(n0)}(0) > (2 \uparrow\uparrow)^{2^{2^{32}+1}-3}2^{2^{2^{32}}}</math>
 '''Upper bound'''
@@ Line 89: / Line 101: @@
 <math>g^{n1}(n0) < (2 \uparrow\uparrow)^{2^{2^{32}+1}-2}8</math>
+or, for a more precise upper bound:
+<math>g^{n1}(n0) < (2 \uparrow\uparrow)^{2^{2^{32}+1}-4}(2^{2^{2^{33}}})</math> and <math>f^{n}(0) < 2 \uparrow\uparrow 3n</math>
+<math>f^{g^{n1}(n0)}(0) < 2 \uparrow\uparrow (3 \times (2 \uparrow\uparrow)^{2^{2^{32}+1}-4}(2^{2^{2^{33}}})) < 2 \uparrow\uparrow (2 \uparrow\uparrow)^{2^{2^{32}+1}-4}(2^{2^{2^{33}}}+1) => (2 \uparrow\uparrow)^{2^{2^{32}+1}-3}(2^{2^{2^{33}}}+1)</math>
+<math>f^{g^{n1}(n0)}(0) < (2 \uparrow\uparrow)^{2^{2^{32}+1}-3}(2^{2^{2^{33}}}+1)</math>
+or, for an even more precise upper bound:
+<math>g^{n1}(n0) < (2 \uparrow\uparrow)^{2^{2^{32}+1}-4}(2^{2^{2^{32}+2}})</math> and <math>f^{n}(0) < 2 \uparrow\uparrow 3n</math>
+<math>f^{g^{n1}(n0)}(0) < 2 \uparrow\uparrow (3 \times (2 \uparrow\uparrow)^{2^{2^{32}+1}-4}(2^{2^{2^{32}+2}})) < 2 \uparrow\uparrow (2 \uparrow\uparrow)^{2^{2^{32}+1}-4}(2^{2^{2^{32}+2}}+1) => (2 \uparrow\uparrow)^{2^{2^{32}+1}-3}(2^{2^{2^{32}+2}}+1)</math>
+<math>f^{g^{n1}(n0)}(0) < (2 \uparrow\uparrow)^{2^{2^{32}+1}-3}(2^{2^{2^{32}+2}}+1)</math>
 ==Lower bound on Σ==
 Σ = <math>5 \times 2^{2^{f^{g^{n1}(n0)}(0)+1}+2}+7 > 5 \times 2^{2^{(2 \uparrow\uparrow\uparrow 2^{2^{32}+1})+1}+2}+7 > 2 \uparrow\uparrow\uparrow 2^{2^{32}+1}</math>
@@ Line 94: / Line 122: @@
 Σ > <math>2 \uparrow\uparrow\uparrow 2^{2^{32}+1}</math>
-or, more precisely:
+or, for a more precise lower bound:
-Σ = <math>5 \times 2^{2^{f^{g^{n1}(n0)}(0)+1}+2}+7 > 2^{2^{(2 \uparrow\uparrow)^{2^{2^{32}+1}-2}6}} > (2 \uparrow\uparrow)^{2^{2^{32}+1}-2}6</math>
+Σ = <math>5 \times 2^{2^{f^{g^{n1}(n0)}(0)+1}+2}+7 > 2^{2^{(2 \uparrow\uparrow)^{2^{2^{32}+1}-3}2^{2^{2^{32}}}}} > (2 \uparrow\uparrow)^{2^{2^{32}+1}-3}2^{2^{2^{32}}}</math>
-Σ > <math>(2 \uparrow\uparrow)^{2^{2^{32}+1}-2}6</math>
+Σ > <math>(2 \uparrow\uparrow)^{2^{2^{32}+1}-3}2^{2^{2^{32}}}</math>
 '''Upper bound'''
@@ Line 107: / Line 135: @@
 Σ < <math>2 \uparrow\uparrow\uparrow (2^{2^{32}+1}+1)</math>
+or, for a more precise upper bound:
+Σ = <math>5 \times 2^{2^{f^{g^{n1}(n0)}(0)+1}+2}+7 < 2^{2^{2^{(2 \uparrow\uparrow)^{2^{2^{32}+1}-3}(2^{2^{2^{33}}}+1)}}} < (2 \uparrow\uparrow)^{2^{2^{32}+1}-3}(2^{2^{2^{33}}}+2)</math>
+Σ < <math>(2 \uparrow\uparrow)^{2^{2^{32}+1}-3}(2^{2^{2^{33}}}+2)</math>
+or, for an even more precise upper bound:
+Σ = <math>5 \times 2^{2^{f^{g^{n1}(n0)}(0)+1}+2}+7 < 2^{2^{2^{(2 \uparrow\uparrow)^{2^{2^{32}+1}-3}(2^{2^{2^{32}+2}}+1)}}} < (2 \uparrow\uparrow)^{2^{2^{32}+1}-3}(2^{2^{2^{32}+2}}+2)</math>
+Σ < <math>(2 \uparrow\uparrow)^{2^{2^{32}+1}-3}(2^{2^{2^{32}+2}}+2)</math>
 =General bound on Σ=
-<math>2 \uparrow\uparrow\uparrow 2^{2^{32}+1}</math> < Σ < <math>2 \uparrow\uparrow\uparrow (2^{2^{32}+1}+1)</math>
+<math>2 \uparrow\uparrow\uparrow 2^{2^{32}+1}</math> < '''Σ''' < <math>2 \uparrow\uparrow\uparrow (2^{2^{32}+1}+1)</math>
+More precisely: <math>(2 \uparrow\uparrow)^{2^{2^{32}+1}-2}6</math> < '''Σ''' < <math>(2 \uparrow\uparrow)^{2^{2^{32}+1}-2}7</math>
-More precisely: <math>(2 \uparrow\uparrow)^{2^{2^{32}+1}-2}6</math> < Σ < <math>(2 \uparrow\uparrow)^{2^{2^{32}+1}-2}9</math>
+Even more precisely: <math>(2 \uparrow\uparrow)^{2^{2^{32}+1}-2}6 < (2 \uparrow\uparrow)^{2^{2^{32}+1}-3}2^{2^{2^{32}}}</math> < '''Σ''' < <math>(2 \uparrow\uparrow)^{2^{2^{32}+1}-3}(2^{2^{2^{32}+2}}+2) < (2 \uparrow\uparrow)^{2^{2^{32}+1}-3}(2^{2^{2^{33}}}+2) < (2 \uparrow\uparrow)^{2^{2^{32}+1}-2}7</math>

User:Polygon/Better lower bound for BB(4,3): Difference between revisions

Latest revision as of 15:08, 18 August 2025

Contents

Better lower bound for BB(4,3)

Definitions

Lower bound on f^n(0)

Lower bound on g^k(n)

Lower bound on g^n1(n0)

Lower bound on f^g^n1(n0)(0)

Lower bound on Σ

General bound on Σ

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