Sum of the reciprocals of all Mersenne prime exponents

[Dobri]

Originally Posted by jnml

Coincidence? I don't think so /s

Well, the comment ends in /s (which means being sarcastic) so one should take it with a pinch of salt.

[Dobri]

Originally Posted by Dobri

An attempt to apply the Wynn's Epsilon Method https://mathworld.wolfram.com/WynnsEpsilonMethod.html for convergence acceleration gives 2.813839133... which is greater than 2.813838890...

Assuming that Sum_EPSILON is close to the unknown Sum_TOTAL, one could entertain the idea of estimating a wide range for M₅₂ as follows:

π(M₅₂) ≈ (1 + π(M₅₂)/π(M₅₃) + π(M₅₂)/π(M₅₄) + ...) / (Sum_EPSILON - Sum₅₁)

for some π(M₅₂)/π(M₅₃) < 1, π(M₅₂)/π(M₅₄) < 1,...

[Dr Sardonicus]

Originally Posted by charybdis

I don't think it's even known that there are infinitely many composite 2^p-1 - which makes proving the infinitude of Mersenne primes seem pretty unrealistic.

The things we don't know about 2^p - 1, p prime, are legion. We don't know whether infinitely many are prime. [If there are only finitely many Mersenne primes, the sum of the reciprocals of the exponents is a finite sum of rational numbers, hence is rational.) We don't know whether infinitely many are composite. (If 2^p - 1 is prime for all but finitely many p, the sum diverges.) Just about everybody and his uncle believes there are infinitely many primes p = 4n+3 for which 8n+ 7 is also prime (though AFAIK nobody knows how to prove it), so "prime for all but finitely many p" seems unlikely. If 2^p - 1 is prime for "enough" p (I think a positive proportion is enough but am not certain) the sum diverges.

We don't even know whether 2^p - 1 is square-free for infinitely many primes p, though AFAIK not a single prime p is known for which 2^p - 1 is divisible by the square of any prime.

[sweety439]

The sequence 2^n-1 can be generalized to any exponential sequence (a*b^n+c)/gcd(a+c,b-1) (with a>=1 is integer, b>=2 is integer, c is (positive or negative) integer, |c|>=1, gcd(a,c) = 1, gcd(b,c) = 1), if this sequence does not have full numerical covering set (e.g. 78557*2^n+1, (11047*3^n+1)/2, (419*4^n+1)/3, 509203*2^n-1, (334*10^n-1)/9, 14*8^n-1, etc.) or full algebraic covering set (e.g. 1*8^n+1, 8*27^n+1, (1*8^n-1)/7, (1*9^n-1)/8, 4*9^n-1, 2500*16^n+1, etc.) or partial algebraic/partial numerical covering set (e.g. 4*24^n-1, 4*39^n-1, (4*19^n-1)/3, (343*10^n-1)/9, 1369*30^n-1, 2500*55^n+1, etc.), we can find the sum of the reciprocals of the positive integers n such that (a*b^n+c)/gcd(a+c,b-1) is prime.

Conjecture: If sequence (a*b^n+c)/gcd(a+c,b-1) does not have covering set (full numerical covering set, full algebraic covering set, or partial algebraic/partial numerical covering set), then the sum of the reciprocals of the positive integers n such that (a*b^n+c)/gcd(a+c,b-1) is prime is converge (i.e. not infinity) and transcendental number. (of course, this conjecture will imply that there are infinitely many such n)

[charybdis]

Originally Posted by sweety439

Conjecture: If sequence (a*b^n+c)/gcd(a+c,b-1) does not have covering set (full numerical covering set, full algebraic covering set, or partial algebraic/partial numerical covering set), then the sum of the reciprocals of the positive integers n such that (a*b^n+c)/gcd(a+c,b-1) is prime is converge (i.e. not infinity) and transcendental number. (of course, this conjecture will imply that there are infinitely many such n)

This is almost certainly false. We expect that for any b there are finitely many primes of the form b^n+1, since these can only be prime when n is a power of 2 (generalized Fermat numbers). The same holds for (b^n+1)/2 when b is odd.

[RomanM]

Originally Posted by Dr Sardonicus

It seems "obvious" that the Mersenne prime exponents (primes p such that 2^p - 1 is prime) are sufficiently rare that the sum of their reciprocals converges. However, I am not aware of any proof of this.

Much better this way, this series converges absolutely) sum = sum + (-1)^n/Mexponent[[n]]

[Dobri]

Originally Posted by RomanM

Much better this way, this series converges absolutely) sum = sum + (-1)^n/Mexponent[[n]]

Perhaps the multiplier (-1)^(n-1) could be used instead for the sum of the alternating sequence to be positive, 0.2697096036...

Also, the sum of the alternating sequence of the reciprocals of the number of primes π(p) less than or equal to the known Mersenne prime exponents tends to converge to 0.6628596647...

[a1call]

I have no way of proving any of the following, so I state them as virtual (as opposed to literal) wagers:

* There are infinite Mersenne primes
* The sum of the reciprocals of Mersenne-Prime-Exponents does not converge

My reasoning for positing The 2nd statement is that for the known Mersenne-Primes the tendency is for the next exponent of each exponent to be less than double.

This would make it less convergent than the infinite series:

(1/2)^1 + (1/2)^2 + (1/2)^3 + .....

Just my 2 cents.

[charybdis]

Originally Posted by a1call

My reasoning for positing The 2nd statement is that for the known Mersenne-Primes the tendency is for the next exponent of each exponent to be less than double.

This would make it less convergent than the infinite series:

(1/2)^1 + (1/2)^2 + (1/2)^3 + .....

What about the sequence 1.1, 1.1^2, 1.1^3, ...? Every term is much less than double the previous term, but the geometric series (1/1.1) + (1/1.1)^2 + (1/1.1)^3 + ... still converges. This is basic high-school stuff.

Or what about square numbers? The ratio between successive squares tends to 1, but the sum of the reciprocals converges.

[a1call]

Originally Posted by charybdis

What about the sequence 1.1, 1.1^2, 1.1^3, ...? Every term is much less than double the previous term, but the geometric series (1/1.1) + (1/1.1)^2 + (1/1.1)^3 + ... still converges. This is basic high-school stuff.

Or what about square numbers? The ratio between successive squares tends to 1, but the sum of the reciprocals converges.

sum_(n=1)^∞ 1.1^n diverges to ∞

https://www.wolframalpha.com/input/?...%5En+converges

[charybdis]

Originally Posted by a1call

sum_(n=1)^∞ 1.1^n diverges to ∞

Well yes, obviously it does. But if you read my post properly you'll see that I was talking about the sum of the reciprocals of that sequence.