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June 20th, 2014, 06:17 AM  #1 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  What's special about 71556319?
1/2 + 1/3 + 1/5 + ... + 1/54617881 + 1/71556319 + 1/640488930211807 + 1/31479896620985421014853629981 + ... = $\pi.$ This is the greedy representation of $\pi$ as a sum of prime reciprocals. The ellipsis represents the 3260801 primes between 5 and 54617881; 71556319 is the first prime outside that block. I considered asking this as an unfair question at the Q&A thread but thought it would be better to post it separately. 
June 21st, 2014, 06:56 AM  #2 
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory 
Just wondering: Does the growth of the denominators in the greedy expansion of some transcendental depends on its transcendence measure?
Last edited by skipjack; November 19th, 2014 at 01:32 AM. 
June 22nd, 2014, 07:50 AM  #3 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms 
I don't think so. You'd expect about that much growth from any number.

June 22nd, 2014, 09:07 AM  #4 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,313 Thanks: 2447 Math Focus: Mainly analysis and algebra 
I still don't see why that should make it special. The block you site looks fairly arbitrary. Are you saying that it's the first nonconsecutive prime reciprocal in the expression?

June 22nd, 2014, 09:45 AM  #5 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  

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