User:Chrisdecorte

Chris De Corte is a freelance consultant living in Aalst (Belgium) and is among others also a mathematical hobbyist.

Chris is interested in unsolved problems

Chris found that the equality between the Riemann zèta function and the Euler product does not seem to hold (for s<=1). This is explained here.

Chris independently developed his own sieve: the "Sine Sieve" which has some resemblance with the Sieve of Eratosthenes and which is explained here.

Chris independently derived 2 new formulas to determine if a given number n is prime or not:

$$P(n) = \prod_{i=2}^{n-1} \sin(\frac{\pi n}{i}) <> 0 \Rightarrow n = \textrm{Prime}$$

and:

$$\textrm{n = Prime}(=p_{i+1}) \iff \textrm{P(n)=} \prod_{p_i=2}^{\forall p_i 0 $$

Using these formulas, he can prove twice Goldbach's Conjecture & Twin prime Conjecture.

This is done by replacing n with 2n=p+q (p and q being 2 primes) and working out the sinus terms.

One of the 2 primes primes that compose the Goldbach requirement needs to be a solution to the following equation:

$$G(x) = \prod_{i=2}^{x-1} \sin(\frac{\pi x}{i}) \cdot \prod_{j=2}^{(2n-x)-1} \sin(\frac{\pi (2n-x)}{j}) <> 0 \Rightarrow x = \textrm{q \quad where \quad 2n=p+q}$$

Other Prime testing formula's he developed are (for those who can't see the beauty of the sine function):

$$P(n) = \prod_{i=1 or 2}^{n-1} GCD(n,i) = 1 \Rightarrow n = \textrm{Prime}$$

and:

$$P(n) = \sum_{i=1}^{n-1} (-1)^i.GCD(n,i) = 0 \Rightarrow n \textrm{\quad could \quad be \quad Prime}$$

This last formula is true for primes but is also true for some non-primes as 4, 9, 15, ...

Chris also developed multiple prime counting formula:

His probabilistic prime counting formula is his final one and can be represented as follows:

$$\pi(x=p_i)=\alpha.\int_2^x\prod_{i=2}^{x=p_i} (1-1/p_{i}).dx \ with \ \alpha\approx1.7810292$$

where $$\alpha$$ can be very closely approximated as:

$$\alpha=e^\gamma \ where \ \gamma\approx0.57721 \ is \ the \ Euler-Mascheroni constant$$

The origin of this formula can be found here and it will take a long time before someone will improve the accurateness of this formula. A video about this formula can be found here.

His previous formula had also very good accuracy:

$$\pi(x)={x \over 2} \cdot {(1- \sqrt{1-{4 \over \ln(x)}})}-7 $$

Others are:

$$\pi(x)=\alpha.x^{\beta} \quad \textrm{with:} \quad \alpha=0.2083666 \quad and \quad \beta=0.9294465$$

This formula seems to be better than the pure version of the Logarithmic Integral x/lnx up to approximately 1E+10 (except for a short range between 2 and 9000).

Therefore, he would like to propose the following improved formula:

$$\pi(x)={\alpha.x^{\beta}.10^{(\gamma-x)}+x/\ln x\over 1+10^{(\gamma-x)}} \quad \textrm{with:} \quad \alpha=0.2083666 \quad \beta=0.9294465 \quad \gamma=1E+10 $$

Other works:

Chris found a very close approximation to the angle trisection problem

He also found an approximation to the squaring of the circle and made some interesting but unanswered comments.

Category:Mathematicians Category:Unsolved problems in mathematics Category:Prime numbers Category:Primality tests‎ Category:Theorems about prime numbers

I calculated the approximation of some famous constants as a fractal.