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 tetration base > exp(2/5) tommy1729 Ultimate Fellow     Posts: 1,370 Threads: 335 Joined: Feb 2009 02/10/2015, 10:43 PM For all clarity the method works for bases larger then eta. But that does not imply x < M(x) < exp(x) where M(x) is a suitable multisection type function. ( Linear combinations of Mittag-Leffler functions ) it is only required for large x that x < M(x) < exp(x). But this follows automatically from the asymptotic behaviour of the multisection type function and the method we use : f(x) ln( f( exp(x) ) ) ... ln^[n]( f ( exp^[n](x) ) ) the third last line implies exp(x) < M(exp(x)) < exp(exp(x)) ... exp^[n](x) < M(exp^[n](x)) < exp^[n+1](x) and hence the method works. However it is intresting to consider exp(x) - M(x). This quantity affects how many iterations we need to take to get a good numeric result. And I find it intresting by itself. The number of sign changes / zero's of exp(x) - M(x) can be estimated by fourier-budan for instance. But there must be many tools for this. And the question of closed form zero's occurs naturally. at x = 1 we can relatively easy check the sign of exp(x) - M(x). M(x) = f(x) = 0 + 5/2 x + 5/2 x^5/5! + 5/2 x^6/6! + 5/2 x^10/10! + 5/2 x^11/11! + ... then exp(1) - M(1) = e - 5/2 (1 + 1/5! + 1/6! + 1/10! +1/11! + ... ) truncated (at ...) this gives us : 0.193976 Notice the number of sign changes of exp(x) - M(x) is Always bounded as function of b in a/b ~ 1/e. More investigation is desired. --- Another thing is the sequence given before 2,5,11,23,47,... these are the iterations by the map 2x+1 starting at 2. This has a closed form : 3 * 2^n - 1. ( trivial to prove ) A similar sequence is 1,7,19,... which are the iterations by the map 2x+5. This also has a closed form : 3 * 2^n - 5. Numbers of the form 3 * 2^n - 1 are called Thâbit ibn Kurrah numbers. and if those numbers are prime they are called Thâbit ibn Kurrah primes. Now if for a fixed n , 3 * 2^n - 1 and 3 * 2^n - 5 are both prime then we have a Thâbit ibn Kurrah cousin prime. Conjecture : there are infinitely many Thâbit ibn Kurrah cousin primes. 7,11 19,23 41,47 379,383 ... regards tommy1729 « Next Oldest | Next Newest »

 Messages In This Thread tetration base > exp(2/5) - by tommy1729 - 02/09/2015, 11:59 PM RE: tetration base > exp(2/5) - by tommy1729 - 02/10/2015, 10:43 PM RE: tetration base > exp(2/5) - by tommy1729 - 02/11/2015, 12:29 AM

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