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March 29th, 2009, 07:25 PM  #1 
Newbie Joined: Mar 2009 Posts: 4 Thanks: 0  Values of a*e+b*pi where a and b are natural numbers
Say one has a ascending list of all values of a*e + b*pi where a and b are natural numbers (0, e, pi, 2e, e+pi, 2pi, 3e, 2e+pi...). Then one looked at the difference of term n and term n1. These differences have very few values: e, 2epi, 3e2pi, 4e3pi, 5e4pi, 6e5pi, pie, 7pi8e, 7e6pi, 22e19pi, 13pi15e, 37e32pi, 89e77pi, 45pi52e, 103pi119e, and 141e122pi account for the first 28,000+ differences between values. Is there any way to predict these differences, or successive ascending values?

March 29th, 2009, 07:38 PM  #2  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Values of a*e+b*pi where a and b are natural numbers Quote:
This actually looks like an interesting and 'deep' question. Looking at the continued fraction expansion of their ratio may give some insight, I'm not sure.  
June 9th, 2009, 01:53 PM  #3  
Member Joined: Jun 2009 Posts: 83 Thanks: 1  Re: Values of a*e+b*pi where a and b are natural numbers Quote:
Or maybe I don't understand ordering and generation of the set of these numbers.  
June 9th, 2009, 08:11 PM  #4  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Values of a*e+b*pi where a and b are natural numbers Quote:
0, e, pi, 2e, e+pi, 2pi, 3e, 2e+pi, taking its first difference, e0, pi  e, 2e  pi, (e + pi)  2e = pi  e, 2pi  (e + pi) = pi  e, 3e  2pi, ..., and removing duplicates: e, 2epi, 3e2pi, 4e3pi, 5e4pi, 6e5pi, pie, 7pi8e, 7e6pi, 22e19pi, 13pi15e, 37e32pi, 89e77pi, 45pi52e, 103pi119e, 141e122pi, ....  
June 10th, 2009, 12:34 PM  #5 
Member Joined: Jun 2009 Posts: 83 Thanks: 1  Re: Values of a*e+b*pi where a and b are natural numbers
At the first sight  to predict the sequnce is I think almost as hard as to predict the next digit in pi expansion.

June 10th, 2009, 06:31 PM  #6  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Values of a*e+b*pi where a and b are natural numbers Quote:
 

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