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Series of exponential prime reciprocals

  1. Feb 27, 2012 #1
    Sum of reciprocal of some base (I just chose e as example) to prime power?

    Ʃ [itex]\frac{1}{e^{p}}[/itex] = [itex]\frac{1}{e^2}[/itex]+[itex]\frac{1}{e^3}[/itex]+[itex]\frac{1}{e^5}[/itex]+[itex]\frac{1}{e^7}[/itex]+[itex]\frac{1}{e^{11}}[/itex]+[itex]\frac{1}{e^{13}}[/itex]+[itex]\frac{1}{e^{17}}[/itex]+...

    Brute force simulation gives me
    Is there an elementary, non-transient solution?
  2. jcsd
  3. Feb 29, 2012 #2
    If you replace e by 2, you get what some people call the prime constant. It encodes all the primes, but as far as I can tell, no other interesting relations involving this number has been found. Also, it may still be open whether this number is algebraic or transcendental.
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