The Ramanujan Machine: Using algorithms to discover new mathematics

Re:The Ramanujan Machine: Using algorithms to discover new m... by klipt on Thursday July 04, 2019 @12:13PM (#20355732)

We can already do this approximately (asymptotically): https://en.wikipedia.org/wiki/Prime_number_theorem

"The prime number theorem is equivalent to the statement that the nth prime number p_n satisfies

p_n ~ nlog(n)

the asymptotic notation meaning, again, that the relative error of this approximation approaches 0 as n increases without bound. For example, the 2^1017th prime number is 8512677386048191063, and (2^1017)log(2^1017) rounds to 7967418752291744388, a relative error of about 6.4%."

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Re:The Ramanujan Machine: Using algorithms to discover new m... by eesmith on Thursday July 04, 2019 @02:47PM (#20356833)

https://www.johndcook.com/blog/2019/06/20/bounds-on-the-nth-... is a recent blog posted titled "Bounds on the nth prime". It lists tighter upper and lower bounds:

    from math import log
    def f(n, k):
      ln = log(n); lln = log(log(n))
      return ln + lln - 1 + (lln-2)/ln - ((lln**2) - 6*lln+k)/(2*ln*ln)

With that in place, and once I realized that 2^1017 mean 2E17 not 2 to the power of 1017:

   >>> n = 2E17
   >>> lo = n * f(n, 11.847)
   >>> hi = n * f(n, 10.273)
   >>> lo
   8.512627944213742e+18
   >>> hi
   8.512727125430618e+18
   >>> exact = 8512677386048191063
   >>> (exact - lo)/exact
   5.80802398678381e-06
   >>> (exact - hi)/exact
   -5.842977499434444e-06

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