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April 9th, 2017, 06:27 AM   #1
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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.
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April 9th, 2017, 11:00 AM   #2
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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!
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April 10th, 2017, 10:28 AM   #3
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x(1000)=1.3957999

What about x(n) when n goes to infinite?
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April 11th, 2017, 04:49 AM   #4
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Conjecture :
As n goes to infinite x(n)=sqrt(2)

If it is true then what?
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April 11th, 2017, 08:53 PM   #5
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Pls rewrite in LaTex or it seems just a stupid question
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