User:Polygon/Better lower bound for BB(4,3) - Revision history

Polygon: Added an even more precise upper bound

2025-08-18T15:08:07Z

Added an even more precise upper bound

← Older revision		Revision as of 15:08, 18 August 2025
Line 71:		Line 71:

	<math>g^{n1}(n0) < (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)==		==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>		<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 102:		Line 108:
	<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 (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^{34}}})</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==		==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>		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 127:		Line 141:

	Σ < <math>(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=		=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}9</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>

	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^{33}}}+2) < (2 \uparrow\uparrow)^{2^{2^{32}+1}-2}7</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>

Polygon: Added a more precise lower bound

2025-08-18T14:45:48Z

Added a more precise lower bound

← Older revision		Revision as of 14:45, 18 August 2025
Line 82:		Line 82:
	or, for a more precise lower bound:		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'''		'''Upper bound'''
Line 108:		Line 108:
	Σ > <math>2 \uparrow\uparrow\uparrow 2^{2^{32}+1}</math>		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'''		'''Upper bound'''
Line 132:		Line 132:
	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>		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</math> < Σ < <math>(2 \uparrow\uparrow)^{2^{2^{32}+1}-3}(2^{2^{2^{33}}}+2) < (2 \uparrow\uparrow)^{2^{2^{32}+1}-2}7</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^{33}}}+2) < (2 \uparrow\uparrow)^{2^{2^{32}+1}-2}7</math>

Polygon: Added more precise upper bounds

2025-08-18T14:22:16Z

Added more precise upper bounds

← Older revision		Revision as of 14:22, 18 August 2025
Line 62:		Line 62:
	'''Upper bound'''		'''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>		<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>
	==Lower bound on f^g^n1(n0)(0)==		==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>		<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 89:		Line 95:

	<math>g^{n1}(n0) < (2 \uparrow\uparrow)^{2^{2^{32}+1}-2}8</math>		<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^{34}}})</math>
	==Lower bound on Σ==		==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>		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 107:		Line 121:

	Σ < <math>2 \uparrow\uparrow\uparrow (2^{2^{32}+1}+1)</math>		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>
	=General bound on Σ=		=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}9</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</math> < Σ < <math>(2 \uparrow\uparrow)^{2^{2^{32}+1}-3}(2^{2^{2^{33}}}+2) < (2 \uparrow\uparrow)^{2^{2^{32}+1}-2}7</math>

Polygon: Added upper bound

2025-08-18T13:30:54Z

Added upper bound

@@ Line 1: / Line 1: @@
 =Better lower bound for BB(4,3)=
 ==Definitions==
@@ Line 20: / Line 21: @@
 <math>=> f^{n}(0) > 2 \uparrow\uparrow 2n</math>
 ==Lower bound on g^k(n)==
 <math>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)</math>
@@ Line 29: / Line 44: @@
 <math>g^{k}(n) > (2 \uparrow\uparrow)^{k}(2n+2) > (2 \uparrow\uparrow)^{k}2n</math>
 ==Lower bound on g^n1(n0)==
 <math>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}}}</math> ; Note that <math>2 \uparrow\uparrow 6 < 2^{2^{2^{32}}} < 2 \uparrow\uparrow 7</math>
@@ Line 35: / Line 59: @@
 <math>g^{n1}(n0) > (2 \uparrow\uparrow)^{2^{2^{32}+1}-2}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 43: / Line 73: @@
 <math>f^{g^{n1}(n0)}(0) > 2 \uparrow\uparrow\uparrow 2^{2^{32}+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>
 Σ > <math>2 \uparrow\uparrow\uparrow 2^{2^{32}+1}</math>

Polygon: Fixed weird headings

2025-08-18T11:51:47Z

Fixed weird headings

← Older revision		Revision as of 11:51, 18 August 2025
Line 10:		Line 10:

	Σ = <math>5 \times 2^{2^{f^{g^{n1}(n0)}(0)+1}+2}+7</math>		Σ = <math>5 \times 2^{2^{f^{g^{n1}(n0)}(0)+1}+2}+7</math>
	==Lower bound on ~~<math>~~f^{n}(0)~~</math>~~==		==Lower bound on f^n(0)==
	<math>f(n) = 2^{2^{n+1}}</math>		<math>f(n) = 2^{2^{n+1}}</math>

Line 20:		Line 20:

	<math>=> f^{n}(0) > 2 \uparrow\uparrow 2n</math>		<math>=> f^{n}(0) > 2 \uparrow\uparrow 2n</math>
	==Lower bound on ~~<math>~~g^{k}(n)~~</math>~~==		==Lower bound on g^k(n)==
	<math>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)</math>		<math>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)</math>

Line 29:		Line 29:
	<math>g^{k}(n) > (2 \uparrow\uparrow)^{k}(2n+2) > (2 \uparrow\uparrow)^{k}2n</math>		<math>g^{k}(n) > (2 \uparrow\uparrow)^{k}(2n+2) > (2 \uparrow\uparrow)^{k}2n</math>

	==Lower bound on ~~<math>~~g^{n1}(n0)~~</math>~~==		==Lower bound on g^n1(n0)==
	<math>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}}}</math> ; Note that <math>2 \uparrow\uparrow 6 < 2^{2^{2^{32}}} < 2 \uparrow\uparrow 7</math>		<math>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}}}</math> ; Note that <math>2 \uparrow\uparrow 6 < 2^{2^{2^{32}}} < 2 \uparrow\uparrow 7</math>

Line 35:		Line 35:

	<math>g^{n1}(n0) > (2 \uparrow\uparrow)^{2^{2^{32}+1}-2}2</math>		<math>g^{n1}(n0) > (2 \uparrow\uparrow)^{2^{2^{32}+1}-2}2</math>
	==Lower bound on ~~<math>~~f^{g^{n1}(n0)}(0)~~</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>		<math>g^{n1}(n0) > (2 \uparrow\uparrow)^{2^{2^{32}+1}-2}2</math> and <math>f^{n}(0) > 2 \uparrow\uparrow 2n</math>

Polygon: Created page for a slightly bigger lower bound for BB(4,3)

2025-08-18T11:50:21Z

Created page for a slightly bigger lower bound for BB(4,3)

New page

=Better lower bound for BB(4,3)=
==Definitions==
<math>f(n) = 2^{2^{n+1}}</math>

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

<math>n0 = \frac {5 \times 2^{2^{2^{32}+1}+1}-4}{9}</math>

<math>n1 = 2^{2^{32}+1}-4</math>

Σ = <math>5 \times 2^{2^{f^{g^{n1}(n0)}(0)+1}+2}+7</math>
==Lower bound on <math>f^{n}(0)</math>==
<math>f(n) = 2^{2^{n+1}}</math>

<math>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</math>

<math>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)</math>

<math>=> f^{k}(0) > (2 \uparrow)^{2k}1 = 2 \uparrow\uparrow 2k</math>

<math>=> f^{n}(0) > 2 \uparrow\uparrow 2n</math>
==Lower bound on <math>g^{k}(n)</math>==
<math>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)</math>

<math>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)</math>

<math>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)</math>

<math>g^{k}(n) > (2 \uparrow\uparrow)^{k}(2n+2) > (2 \uparrow\uparrow)^{k}2n</math>

==Lower bound on <math>g^{n1}(n0)</math>==
<math>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}}}</math> ; Note that <math>2 \uparrow\uparrow 6 < 2^{2^{2^{32}}} < 2 \uparrow\uparrow 7</math>

<math>(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</math>

<math>g^{n1}(n0) > (2 \uparrow\uparrow)^{2^{2^{32}+1}-2}2</math>
==Lower bound on <math>f^{g^{n1}(n0)}(0)</math>==
<math>g^{n1}(n0) > (2 \uparrow\uparrow)^{2^{2^{32}+1}-2}2</math> and <math>f^{n}(0) > 2 \uparrow\uparrow 2n</math>

<math>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</math>

<math>(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}</math>

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

Σ > <math>2 \uparrow\uparrow\uparrow 2^{2^{32}+1}</math>