3
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1: Prime gap bounds:

Consider the following non-asymptotic bounds for $\pi(x)$, proven by Dusart in 2018 (holding for $x>5393$):

$$\frac{x}{\log(x)-1}<\pi(x)<\frac{x}{\log(x)-1.112}$$

To get the maximum value for $p(n)$, we take the lower bound with $x=p(n)$ :

$$\pi(p(n))=\frac{p(n)}{\log p(n)-1}$$ $$n\left(\log p(n)-1\right)-p(n)=0$$ $$p(n)=-nW_{-1}\left(\frac{-e^{1}}{n}\right)$$

$W_{k}(n)$ is the analytic continuation of the product log function. Now if we do the same with the upper bound, we get non-asymptotic bounds for the $N^{th}$ prime $p(n)$:

$$-nW_{-1}\left(\frac{-e^{1.112}}{n}\right)<p(n)<-nW_{-1}\left(\frac{-e}{n}\right)$$

Using our bounds for $p(n)$, we can get a upper bound for the $n^{th}$ prime gap $g(n)$. Since $g(n)=p(n+1)-p(n)$, we will take the upper bound for $p(n+1)$ and the lower bound for $p(n)$:

$$2 \leq g(n)< \left(-(n+1)W_{-1}\left(\frac{-e}{(n+1)}\right)\right)-\left(-nW_{-1}\left(\frac{-e^{1.112}}{n}\right)\right)$$

2: Goldbach critical prime gap:

Let's define $a(n)$, the highest even integer such that all positive even integer $\leq a(n)$ except $2$ and $4$ can be written as the sum of $2$ odd prime $\leq p(n)$. We can now reformulate Goldbach's conjecture as:

$$a(n)\geq p(n), 1<n<\infty$$

Now since we know all even integer from $p(n)$ to $a(n)$ already have at least $1$ Goldbach partition, we can define $c(n)=a(n)-p(n)$ as the Goldbach critical prime gap, i.e the minimum value of the prime gap $g(n)$ such that an even integer between $a(n)$ and $p(n+1)$ have no Goldbach partition.

3: Observations:

Two observations that seems to makes it impossible for Goldbach's conjecture to be false are:

$$\lim_{n\rightarrow\infty}\frac{p(n)}{a(n)}\approx\frac{1}{2}$$

$2^{1}$ | $0.50000$      $2^{10}$ | $0.51310$
$2^{2}$ | $0.53846$      $2^{11}$ | $0.50561$
$2^{3}$ | $0.51351$      $2^{12}$ | $0.50168$
$2^{4}$ | $0.59550$      $2^{13}$ | $0.50159$
$2^{5}$ | $0.54811$      $2^{14}$ | $0.50102$
$2^{6}$ | $0.54657$      $2^{15}$ | $0.50052$
$2^{7}$ | $0.53062$      $2^{16}$ | $0.50031$
$2^{8}$ | $0.52208$      $2^{17}$ | $0.50015$
$2^{9}$ | $0.51711$      $2^{18}$ | $0.50009$

$$\lim_{n\rightarrow\infty}\frac{\left(-(n+1)W_{-1}\left(\frac{-e}{(n+1)}\right)\right)-\left(-nW_{-1}\left(\frac{-e^{1.112}}{n}\right)\right)}{c(n)}\approx0.112$$

$2^{1}$ | $0.00000$      $2^{10}$ | $0.13222$
$2^{2}$ | $1.93764$      $2^{11}$ | $0.12654$
$2^{3}$ | $0.43229$      $2^{12}$ | $0.12335$
$2^{4}$ | $0.37365$      $2^{13}$ | $0.12240$
$2^{5}$ | $0.22899$      $2^{14}$ | $0.12142$
$2^{6}$ | $0.18954$      $2^{15}$ | $0.12059$
$2^{7}$ | $0.15974$      $2^{16}$ | $0.12000$
$2^{8}$ | $0.14525$      $2^{17}$ | $0.11950$
$2^{9}$ | $0.13731$      $2^{18}$ | $0.11909$

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  • 5
    $\begingroup$ "Why can't we prove Goldbach's conjecture with this method?" No one says we can't, but making observations is much harder than proving something. $\endgroup$ – Wojowu Jul 13 '18 at 11:24
  • 6
    $\begingroup$ And also, even if the limit holds, this doesn't mean that there isn't finitely many integers for which the property $a(n) \geq p(n)$ fails to hold $\endgroup$ – asdf Jul 13 '18 at 11:27
  • $\begingroup$ The statistical evidence of the truth of Goldbach's conjecture is (in contrary to Collatz's conjecture) in fact overwhelming, but counterexamples cannot be ruled out this way. A good examples that oberservations can fail are the Goodstein-sequences. If we would let run a computer , we would come to the conclusion that they will never terminate, which is however false. $\endgroup$ – Peter Jul 13 '18 at 11:31
  • $\begingroup$ The twin-prime conjecture is a very similar case. It also seems to be "obviously true", but no proof has been found. There is a formular allowing to count the number of twin-prime-pairs below some $x$, but this only an empirical formula , it does not prove the twin-prime conjecture. $\endgroup$ – Peter Jul 13 '18 at 11:36
  • 1
    $\begingroup$ Even if you had $${x\over\log x-1}<\pi(x)<{x\over \log x-1.000000001}$$ you still couldn't prove Goldbach . $\endgroup$ – Gerry Myerson Jul 13 '18 at 23:33

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