f ( n ) = 2 2 n + 1 {\displaystyle f(n)=2^{2^{n+1}}}
g ( n ) = 5 × 2 2 f n ( 0 ) + 1 + 2 − 8 9 {\displaystyle g(n)={\frac {5\times 2^{2^{f^{n}(0)+1}+2}-8}{9}}}
n 0 = 5 × 2 2 2 32 + 1 + 1 − 4 9 {\displaystyle n0={\frac {5\times 2^{2^{2^{32}+1}+1}-4}{9}}}
n 1 = 2 2 32 + 1 − 4 {\displaystyle n1=2^{2^{32}+1}-4}
Σ = 5 × 2 2 f g n 1 ( n 0 ) ( 0 ) + 1 + 2 + 7 {\displaystyle 5\times 2^{2^{f^{g^{n1}(n0)}(0)+1}+2}+7}
f ( 0 ) = 2 2 1 ; f ( f ( 0 ) ) = 2 2 ( 2 2 1 + 1 ) > ( 2 ↑ ) 4 1 ; f 3 ( 0 ) = 2 2 ( 2 2 ( 2 2 1 + 1 ) + 1 ) > ( 2 ↑ ) 6 1 {\displaystyle 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 ↑ ) 2 f k − 1 ( 0 ) => ( 2 ↑ ) 2 a f k − a ( 0 ) {\displaystyle 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 ↑ ) 2 k 1 = 2 ↑↑ 2 k {\displaystyle =>f^{k}(0)>(2\uparrow )^{2k}1=2\uparrow \uparrow 2k}
=> f n ( 0 ) > 2 ↑↑ 2 n {\displaystyle =>f^{n}(0)>2\uparrow \uparrow 2n}
g ( n ) = 5 × 2 2 f n ( 0 ) + 1 + 2 − 8 9 > 5 × 2 2 2 ↑↑ 2 n + 2 − 8 9 > 2 2 2 ↑↑ 2 n = 2 ↑↑ ( 2 n + 2 ) {\displaystyle 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 ) > 5 × 2 2 f 2 ↑↑ ( 2 n + 2 ) ( 0 ) + 1 + 2 − 8 9 > 5 × 2 2 2 ↑↑ 2 ↑↑ ( 2 n + 2 ) + 2 − 8 9 > 2 ↑↑ 2 ↑↑ ( 2 n + 2 ) {\displaystyle 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 ↑↑ ( 2 × g k − 1 ( n ) + 2 ) = ( 2 ↑↑ ) 1 ( 2 × g k − 1 ( n ) + 2 ) =>> ( 2 ↑↑ ) a ( 2 × g k − a ( n ) + 2 ) {\displaystyle 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 ↑↑ ) k ( 2 n + 2 ) > ( 2 ↑↑ ) k 2 n {\displaystyle g^{k}(n)>(2\uparrow \uparrow )^{k}(2n+2)>(2\uparrow \uparrow )^{k}2n}
g n 1 ( n 0 ) = g 2 2 32 + 1 − 4 ( 5 × 2 2 2 32 + 1 + 1 − 4 9 ) > g 2 2 32 + 1 − 4 ( 2 2 2 32 ) > ( 2 ↑↑ ) 2 2 32 + 1 − 4 2 2 2 32 {\displaystyle 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 ↑↑ 6 < 2 2 2 32 < 2 ↑↑ 7 {\displaystyle 2\uparrow \uparrow 6<2^{2^{2^{32}}}<2\uparrow \uparrow 7}
( 2 ↑↑ ) 2 2 32 + 1 − 4 2 2 2 32 > ( 2 ↑↑ ) 2 2 32 + 1 − 4 2 ↑↑ 6 => ( 2 ↑↑ ) 2 2 32 + 1 − 3 6 > ( 2 ↑↑ ) 2 2 32 + 1 − 3 4 => ( 2 ↑↑ ) 2 2 32 + 1 − 3 2 ↑↑ 2 => ( 2 ↑↑ ) 2 2 32 + 1 − 2 2 {\displaystyle (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 n 1 ( n 0 ) > ( 2 ↑↑ ) 2 2 32 + 1 − 2 2 {\displaystyle g^{n1}(n0)>(2\uparrow \uparrow )^{2^{2^{32}+1}-2}2}
g n 1 ( n 0 ) > ( 2 ↑↑ ) 2 2 32 + 1 − 2 2 {\displaystyle g^{n1}(n0)>(2\uparrow \uparrow )^{2^{2^{32}+1}-2}2} and f n ( 0 ) > 2 ↑↑ 2 n {\displaystyle f^{n}(0)>2\uparrow \uparrow 2n}
f g n 1 ( n 0 ) ( 0 ) > f ( 2 ↑↑ ) 2 2 32 + 1 − 2 2 ( 0 ) > 2 ↑↑ ( 2 × ( 2 ↑↑ ) 2 2 32 + 1 − 2 2 ) > 2 ↑↑ ( 2 ↑↑ ) 2 2 32 + 1 − 2 2 => ( 2 ↑↑ ) 2 2 32 + 1 − 1 2 {\displaystyle 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 ↑↑ ) 2 2 32 + 1 − 1 2 => ( 2 ↑↑ ) 2 2 32 + 1 − 1 2 ↑↑ 1 => ( 2 ↑↑ ) 2 2 32 + 1 1 => 2 ↑↑↑ 2 2 32 + 1 {\displaystyle (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 n 1 ( n 0 ) ( 0 ) > 2 ↑↑↑ 2 2 32 + 1 {\displaystyle f^{g^{n1}(n0)}(0)>2\uparrow \uparrow \uparrow 2^{2^{32}+1}}
Σ = 5 × 2 2 f g n 1 ( n 0 ) ( 0 ) + 1 + 2 + 7 > 5 × 2 2 ( 2 ↑↑↑ 2 2 32 + 1 ) + 1 + 2 + 7 > 2 ↑↑↑ 2 2 32 + 1 {\displaystyle 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 ↑↑↑ 2 2 32 + 1 {\displaystyle 2\uparrow \uparrow \uparrow 2^{2^{32}+1}}
; Note that 
and 
