April 9th, 2017, 06:27 AM  #1 
Senior Member Joined: Dec 2013 Posts: 1,050 Thanks: 31  Set of problems for fun
Hi, Let p and q be prime numbers >=2 (q>p) Is there a way other than some program to solve : (1/p)^x + (1/q)^x=1 More generally is there a way to solve : sum((1/(p(i))^x(i))=1 with varying from 1 to n where p(i) is the ith prime x(i) is the ith variable solving the equation above. Example n=4 (1/2)^x(4) + (1/3)^x(4) + (1/5)^x(4) + (1/7)^x(4) = 1 Last questions : As n goes to infinite what is the limit of x(n)? To which set of numbers belongs x(i)? Thank you. 
April 9th, 2017, 11:00 AM  #2 
Senior Member Joined: Dec 2013 Posts: 1,050 Thanks: 31 
Can you list the first 100 values of x(i) with a format x,xxxxxxxxxx (10 digits after the comma)? You have an infinite sequence of irrational numbers (x(i)) and there is (maybe?) a way to discover a closed formula for x(i). x(1)=0 x(2)=0.7878 (approximation) etc... Good luck! and good work! 
April 10th, 2017, 10:28 AM  #3 
Senior Member Joined: Dec 2013 Posts: 1,050 Thanks: 31 
x(1000)=1.3957999 What about x(n) when n goes to infinite? 
April 11th, 2017, 04:49 AM  #4 
Senior Member Joined: Dec 2013 Posts: 1,050 Thanks: 31 
Conjecture : As n goes to infinite x(n)=sqrt(2) If it is true then what? 
April 11th, 2017, 08:53 PM  #5 
Senior Member Joined: Dec 2012 Posts: 925 Thanks: 23 
Pls rewrite in LaTex or it seems just a stupid question


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