05/23/2014, 11:12 PM
More precisely :
Uniqueness or at least a strong condition :
Let B2 > B1 > e^(1/e).
Base independance :
r > 0
Define for base C :
exp_C(z) = c^z
a [r]_C b = b [r]_C a = exp_C^[r]( exp_C^[-r](a) + exp_C^[-r](b) )
Conjecture :
a [r]_B2 b = b [r]_B2 a = exp_B2^[r]( exp_B2^[-r](a) + exp_B2^[-r](b) ) = a [r]_B1 b = b [r]_B1 a = exp_B1^[r]( exp_B1^[-r](a) + exp_B1^[-r](b) )
Or simply : a [r]_B2 b = a [r]_B1 b.
Remark : its known to be true for r a positive integer. (it fails for r = -1 though )
Unfortunately I lost some data concerning this idea.
So if you see it mentioned here before , plz let me know.
I am also intrested in other " base independance " equations.
https://sites.google.com/site/tommy1729/...e-property
regards
tommy1729
Uniqueness or at least a strong condition :
Let B2 > B1 > e^(1/e).
Base independance :
r > 0
Define for base C :
exp_C(z) = c^z
a [r]_C b = b [r]_C a = exp_C^[r]( exp_C^[-r](a) + exp_C^[-r](b) )
Conjecture :
a [r]_B2 b = b [r]_B2 a = exp_B2^[r]( exp_B2^[-r](a) + exp_B2^[-r](b) ) = a [r]_B1 b = b [r]_B1 a = exp_B1^[r]( exp_B1^[-r](a) + exp_B1^[-r](b) )
Or simply : a [r]_B2 b = a [r]_B1 b.
Remark : its known to be true for r a positive integer. (it fails for r = -1 though )
Unfortunately I lost some data concerning this idea.
So if you see it mentioned here before , plz let me know.
I am also intrested in other " base independance " equations.
https://sites.google.com/site/tommy1729/...e-property
regards
tommy1729