How many primes are there in a certain range of numbers?

  • Context: Graduate 
  • Thread starter Thread starter newcool
  • Start date Start date
  • Tags Tags
    Numbers Primes Range
Click For Summary
SUMMARY

The discussion focuses on estimating the number of prime numbers between 10100 and 10101. The primary method suggested is using the logarithmic integral function, Li(x), defined as Li(x) = ∫0x (dt/log(t)). By calculating Li(10101) and Li(10100) and subtracting the two results, one can obtain a reliable estimate of the prime count in that range. The mention of Riemann's work indicates a potential for more advanced methods, but Gauss's formula remains a practical approach for this estimation.

PREREQUISITES
  • Understanding of prime number theory
  • Familiarity with the logarithmic integral function
  • Basic calculus for evaluating integrals
  • Knowledge of Riemann Hypothesis implications
NEXT STEPS
  • Research the properties of the logarithmic integral function Li(x)
  • Study the Riemann Hypothesis and its significance in prime number distribution
  • Explore numerical methods for computing integrals
  • Learn about advanced prime counting functions beyond Gauss's approximation
USEFUL FOR

Mathematicians, number theorists, and anyone interested in prime number distribution and estimation techniques.

newcool
Messages
42
Reaction score
0
Hi, what would be the best estimate in the # of primes between [tex]10^{100}[/tex] and [tex]10^{101}[/tex]

thanks
 
Physics news on Phys.org
I'm not sure if there is a newer equation, there probably is one from riemman, but Gauss had a formula for approximating the number of primes up to any number x:

[tex]Li(x)=\int_0^x\frac{dt}{log(t)}[/tex]

You could compute this for 10100 and then for 10101 and subtract the first result from the second and it will give a good estimate.
 

Similar threads

Python Who can find the largest prime number with their own programmed code?
  • · Replies 28 ·
Replies
28
Views
6K
MHB Aux24 probability student range
  • · Replies 5 ·
Replies
5
Views
2K
MHB How Can I Solve a Problem Using Euler's Totient Function for Odd Prime Numbers?
  • · Replies 1 ·
Replies
1
Views
1K
MHB Next set of prime birthdays for three brothers
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Discussion on Astronomical Prime Numbers and Re-evaluating the Primali
  • · Replies 5 ·
Replies
5
Views
2K
New Idea for Prime Number Sieve
  • · Replies 36 ·
2
Replies
36
Views
3K
Undergrad Infinitely many primes of the form p² + nq² -- really?
  • · Replies 8 ·
Replies
8
Views
3K
Undergrad I finding the EV of the possible outcomes
  • · Replies 4 ·
Replies
4
Views
1K
Graduate Is it required to use the Reimann function to solve the problem?
  • · Replies 8 ·
Replies
8
Views
2K
MHB How to find whether a given number is prime or not?
  • · Replies 1 ·
Replies
1
Views
2K