Ponder This Challenge - September 2014 - 2^3^4^5 leading digit on base e

Ponder This Challenge:

Let b = (2^3^4^5) / (e^n), otherwise stated as "two to the power of (three to the power of (4 to the power of 5)) over e to the nth power", where n is an integer such that

1 < b < e.

Find b, with an accuracy of 10 decimal digits.

Update (9/11): There is a rather elegant way to solve this problem. We've added a star to solvers who found it.

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Solution

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  ln(2^3^4^5)=3^1024 * ln(2)

  We need to compute the fraction part of 3^1024 * ln(2).

  The Taylor series of ln(x) around 1 converges slowly, so to compute ln(2) one can compute ln(1/2) = -ln(2), which converges much faster; but to earn a star we use the following relation:

  ln(2) = ln(4/3)-ln(2/3)

  which gives us the series

  ln(2) = 2*sum_{i_0}^\infty 3^{-2i-1}\over(2i+1}

  3^1024 has 489 digits, so we need to compute ln(2) with ~500 digits accuracy, but using the above formula we can see that ~550 terms would suffice, since the contribution of the rest to 3^1024 ln(2) is much smaller than 10^-10. So we get that the fractional part is: 0.4261674320657868...

  The answer is the exponent of the above: 1.53137715531...

Solvers

• Dan Dima (08/29/2014 03:57 PM EDT)
• Dezhi Zhao (08/31/2014 04:42 AM EDT)
• *Radu-Alexandru Todor (09/01/2014 07:27 AM EDT)
• Daniel Bitin (08/31/2014 10:04 AM EDT)
• Joaquim Neves Carrapa (08/31/2014 11:31 AM EDT)
• Guangzhong Sun (09/01/2014 10:08 AM EDT)
• avid Brink (09/01/2014 10:50 AM EDT)
• Andreas Stiller (09/01/2014 12:51 PM EDT)
• *Oleg Vlasii (09/01/2014 02:42 PM EDT)
• Albert Stadler (09/01/2014 02:42 PM EDT)
• Sumeet Katariya (09/01/2014 04:19 PM EDT)
• Florian Fischer (EDT)
• James Dow Allen (09/01/2014 08:33 PM EDT)
• Joseph DeVincentis (09/01/2014 09:16 PM EDT)
• John Snyder (09/01/2014 09:29 PM EDT)
• Allen Sirolly (09/01/2014 09:41 PM EDT)
• Vladimir Milovanovic (09/01/2014 11:58 PM EDT)
• Harold Gutch (09/02/2014 12:04 AM EDT)
• Addison Fischer (09/02/2014 12:43 AM EDT)
• Reiner Martin (09/02/2014 01:18 AM EDT)
• Robbie Gibson (09/02/2014 07:38 AM EDT)
• Benjamin Phillabaum (09/02/2014 07:50 AM EDT)
• Arthur Breitman (09/02/2014 07:55 AM EDT)
• José Eduardo Gaboardi de Carvalho (09/02/2014 09:37 AM EDT)
• Armin Krauss (09/02/2014 09:56 AM EDT)
• *Aviv Nisgav (09/02/2014 10:00 AM EDT)
• Gilles-Philippe Paillé (09/02/2014 10:52 AM EDT)
• Serge Batalov (09/02/2014 12:14 PM EDT)
• David Dodson & Donald Dodson (09/02/2014 12:43 PM EDT)
• Sean Egan (09/02/2014 12:49 PM EDT)
• John Tromp (09/02/2014 03:15 PM EDT)
• Hugo Pfoertner (09/02/2014 03:21 PM EDT)
• Mathias Schenker (09/02/2014 03:26 PM EDT)
• Todd Will (09/02/2014 03:29 PM EDT)
• Lorenz Reichel (09/02/2014 03:56 PM EDT)
• Tim Lewis (09/02/2014 05:37 PM EDT)
• Andrea Andenna (09/02/2014 05:50 PM EDT)
• Antoine Comeau (09/02/2014 07:15 PM EDT)
• Ellen Liu (09/02/2014 08:19 PM EDT)
• Olivier Mercier (09/02/2014 08:25 PM EDT)
• Mark Mixer (09/02/2014 08:46 PM EDT)
• Rogerio Ponce da Silva (09/02/2014 09:29 PM EDT)
• Chris Shannon (09/02/2014 11:50 PM EDT)
• Liubing Yu (09/02/2014 11:34 PM EDT)
• Harald Bögeholz (09/03/2014 01:47 AM EDT)
• Hyung Sik Hwang (09/03/2014 02:53 AM EDT)
• Richard Gosiorovsky (09/03/2014 03:15 AM EDT)
• Gurvan Lullien (09/03/2014 04:08 AM EDT)
• Tamir Ganor & Shouky Dan (09/03/2014 08:02 AM EDT)
• Paulo Sousa (09/03/2014 08:58 AM EDT)
• Zhao Yu Dong (09/03/2014 09:40 AM EDT)
• Philip Kinlen (09/03/2014 10:17 AM EDT)
• Vladimir Sedach (09/03/2014 01:40 PM EDT)
• Tim Joseph Clark (09/03/2014 03:10 PM EDT)
• Eliott Suits (09/03/2014 04:03 PM EDT)
• Cynthia Beauchemin (09/03/2014 04:57 PM EDT)
• David Greer (09/03/2014 06:56 PM EDT)
• Tanner Swett (09/03/2014 08:37 PM EDT)
• Abhinav Kumar (09/03/2014 10:40 PM EDT)
• Hannes Schenck (09/04/2014 07:26 AM EDT)
• Stéphane Higueret (09/04/2014 08:39 AM EDT)
• Benjamin Brumfield (09/04/2014 03:53 PM EDT)
• Jong Wook Kim (09/04/2014 04:28 PM EDT)
• Kan Shen (09/04/2014 05:30 PM EDT)
• *Viktor Chayko & Aliaksei Vaidzelevich (09/05/2014 02:12 AM EDT)
• Graham Hemsley (09/05/2014 08:29 AM EDT)
• Amedeo Stracqualursi (09/05/2014 09:53 AM EDT)
• Victor Chang (09/05/2014 06:52 PM EDT)
• Roger Duthie (09/06/2014 06:12 AM EDT)
• Martin Thiim (09/06/2014 07:04 AM EDT)
• Ranjith Gaddam & Mahesh Dhabade (09/06/2014 09:30 AM EDT)
• Anoop Ghanwani (09/07/2014 09:09 AM EDT)
• Alex Schuy (09/07/2014 10:56 AM EDT)
• Motty Porat (09/07/2014 02:02 PM EDT)
• Antonio García Astudillo (09/07/2014 08:21 PM EDT)
• Ken Meehan (09/07/2014 09:05 PM EDT)
• William Shaw (09/08/2014 02:51 AM EDT)
• Peter Gerritson (09/08/2014 06:41 PM EDT)
• Ricky Zhou (09/08/2014 05:14 PM EDT)
• Arthur Vause (09/09/2014 02:06 AM EDT)
• Christian Blatter (09/09/2014 03:09 AM EDT)
• *Misha Berlinkov (09/09/2014 03:23 AM EDT)
• Dieter Beckerle (09/09/2014 10:29 AM EDT)
• Hakan Summakoglu (09/09/2014 01:16 PM EDT)
• Hamidreza Bidar (09/09/2014 09:14 PM EDT)
• Emir Haleva (09/10/2014 04:04 AM EDT)
• Emilio Del Tessandoro (09/10/2014 07:22 AM EDT)
• *Guillaume Bioche (09/10/2014 04:36 PM EDT)
• Eric Ng (09/11/2014 10:28 AM EDT)
• Anders Bisbjerg Madsen (09/11/2014 01:39 PM EDT)
• Michel Hannoyer (09/11/2014 02:47 PM EDT)
• Michael Liepelt (09/12/2014 11:51 AM EDT)
• *Heyan Wu (09/14/2014 12:19 AM EDT)
• Sergey Koposov (09/14/2014 11:42 AM EDT)
• Christopher Edward Marks (09/14/2014 12:10 PM EDT)
• Joe Klobusicky (09/14/2014 03:14 PM EDT)
• Ricardas Svarplis (09/15/2014 02:56 AM EDT)
• Daniel Suen (09/15/2014 03:41 PM EDT)
• Gil Citro (09/15/2014 03:48 PM EDT)
• Roll (09/16/2014 08:45 AM EDT)
• Huikang Yi (09/16/2014 09:12 AM EDT)
• *Harald Bögeholz (09/16/2014 10:47 AM EDT)
• Vanio Beccaccioli (09/16/2014 01:52 PM EDT)
• Frank Schoeps (09/16/2014 02:03 PM EDT)
• Alessandro Marzo (09/16/2014 04:31 PM EDT)
• Ross Lippert (09/17/2014 12:51 PM EDT)
• Don Ryan (09/17/2014 09:39 PM EDT)
• Koszó Norbert (09/18/2014 04:35 AM EDT)
• *Ehud Schreiber (09/18/2014 08:10 AM EDT)
• Harish Cherukuri (09/19/2014 05:43 AM EDT)
• Tim Nordenfur (09/19/2014 07:39 AM EDT)
• Jan Fricke (09/19/2014 09:13 PM EDT)
• *Yasodhar Patnaik (09/19/2014 09:29 PM EDT)
• Michael Rosola (09/20/2014 09:46 AM EDT)
• Boris Rakus (09/20/2014 11:57 AM EDT)
• *Paul McKenney (09/20/2014 11:58 AM EDT)
• William Kang (09/21/2014 09:54 AM EDT)
• Jose Coelho (09/21/2014 12:07 PM EDT)
• Kipp Johnson (09/21/2014 04:53 PM EDT)
• Jaesung Son (09/21/2014 07:39 PM EDT)
• Daniel König (09/23/2014 10:02 AM EDT)
• *Ali Khodabakhsh (09/23/2014 10:32 AM EDT)
• Bradley Sherman (09/23/2014 11:19 AM EDT)
• Yoav Kaempfer (09/23/2014 04:19 PM EDT)
• Phil Benamy (09/23/2014 10:58 PM EDT)
• Denys Kopiychenko (09/24/2014 05:34 AM EDT)
• Gale Greenlee (09/24/2014 01:54 PM EDT)
• Erik Hostens (09/26/2014 01:45 AM EDT)
• Eric Harley (09/26/2014 08:40 AM EDT)
• William Jung (09/29/2014 07:46 PM EDT)
• Per Lundberg (09/30/2014 04:38 AM EDT)
• *David Vainapel (09/27/2014 07:25 AM EDT)
• Bernard Coudry (09/28/2014 03:59 AM EDT)
• June Kang (09/28/2014 10:54 AM EDT)
• Seung Hwan (Sonny) An (09/28/2014 11:11 AM EDT)
• SeongKweon Hong (09/28/2014 01:16 PM EDT)
• Carrie Oh (09/28/2014 04:44 PM EDT)
• Kwonil Kobe Ko (09/28/2014 05:25 PM EDT)
• Oskar Sigvardsson (09/29/2014 11:51 AM EDT)
• Angel Mancebo (09/30/2014 03:14 PM EDT)
• Konstantin Dzhigardzhyan (09/30/2014 03:57 PM EDT)
• Karim Djaidja (10/01/2014 07:15 AM EDT)