Karim Belabas on Sun, 24 Mar 2024 09:53:56 +0100


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Re: Question on PARI/GP realprecision 

* hermann@stamm-wilbrandt.de [2024-03-24 01:43]:
> I have not worked with realprecision until now.
> 
> In
> https://oeis.org/A080076
> 
> it is stated that reciprocal Proth prime sum is in an interval:
> 
> Sum_{n>=1} 1/a(n) is in the interval (0.7473924793, 0.7473924795) (Borsos et
> al., 2022). - Amiram Eldar, Jan 29 2022
[...]
> Biggest Proth prime P[122742] is:
> 
> ? P=readvec("a080076.json.txt")[1];
> ? #P
> 122742
> ? P[#P]
> 1099489607681
> ? vecmax(P)
> 1099489607681
> ?
> 
> 
> What PARI/GP realprecision is needed to correctly deal with this finite
> reciprocal sum?

It already correctly deals with your sum. The result you quote states
that the infinite sum lies in some interval. Not that some random finite
subsum does.

Each (positive!) term in the range you investigate provides a little
less than 1e-12 to the sum (with diminishing returns). You're missing
about 1e-7 to get into the expected range, so you need to add at least
10^5 more primes ...

If you want to convince yourself that precision is not the culprit,
compute the sum of the 1/p, instead of 1.0/p. You'll get a *very* large
rational number, exactly equal to the sum (this is a very wasteful way
to compute, as the denominator is going to be the product of all primes
in the list, but it's a doable calculation).

You will then be able to check that this (exactly known) number does not
belong to your expected interval.

Cheers,

    K.B.
-- 
Pr. Karim Belabas, U. Bordeaux, Vice-président en charge du Numérique
Institut de Mathématiques de Bordeaux UMR 5251 - (+33) 05 40 00 29 77
http://www.math.u-bordeaux.fr/~kbelabas/