Series of exponential prime reciprocals

In summary, a series of exponential prime reciprocals is a mathematical sequence where each term is the reciprocal of a prime number raised to an increasing power. Studying this series can provide insights into prime numbers and has applications in cryptography. It differs from a regular series in terms of its pattern and the behavior of its terms. The formula for finding the nth term is 1/(p^n), where p is the nth prime number. This series can be infinite, but its terms approach 0 as the power increases, resulting in convergence rather than divergence.
  • #1
YvesSch
4
0
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]+...
p[itex]\in[/itex]P

Brute force simulation gives me
~0.19279118970439518
Is there an elementary, non-transient solution?
 
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  • #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.
 

FAQ: Series of exponential prime reciprocals

1. What is a series of exponential prime reciprocals?

A series of exponential prime reciprocals is a mathematical sequence in which each term is the reciprocal of a prime number raised to an increasing power. For example, 1/2, 1/4, 1/8, 1/16, ... is a series of exponential prime reciprocals where each term is the reciprocal of 2 raised to an increasing power.

2. What is the significance of studying series of exponential prime reciprocals?

Studying series of exponential prime reciprocals can provide insights into the distribution and properties of prime numbers. It can also be used in cryptography and other areas of mathematics.

3. How is a series of exponential prime reciprocals different from a regular series?

A regular series follows a specific pattern or rule, whereas a series of exponential prime reciprocals follows a pattern based on prime numbers and their reciprocals. Additionally, as the power increases, the terms in a series of exponential prime reciprocals approach 0, whereas in a regular series, the terms may increase or decrease depending on the pattern.

4. What is the formula for finding the nth term in a series of exponential prime reciprocals?

The formula for finding the nth term in a series of exponential prime reciprocals is 1/(pn), where p is the nth prime number. For example, the 5th term in a series of exponential prime reciprocals would be 1/(55) = 1/3125.

5. Can a series of exponential prime reciprocals be infinite?

Yes, a series of exponential prime reciprocals can be infinite. This is because there are infinitely many prime numbers, and therefore, infinitely many terms in the series. However, as the power increases, the terms approach 0 and become increasingly smaller, making the series converge to a finite value rather than diverging to infinity.

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