User:Chrisdecorte

[http://be.linkedin.com/in/chrisdecorte/ Chris De Corte] is a freelance consultant living in [http://www.aalst.be/ Aalst] (Belgium) and is among others also a mathematical hobbyist.

Chris is interested in [http://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics 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 [http://www.slideshare.net/ChrisDeCorte1/disprove-of-equality-between-riemann-zeta-function-and-euler-product here].

Chris independently developed his own sieve: the "Sine Sieve" which has some resemblance with the Sieve of Eratosthenes and which is explained [http://www.youtube.com/watch?v=ajmvOLrpEAY 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 [http://www.slideshare.net/ChrisDeCorte1/derivation-of-a-prime-verification-formula-to-prove-the-related-open-problems prove] [http://www.slideshare.net/ChrisDeCorte1/new-prime-formula-to-prove-open-problems-ii twice] [http://en.wikipedia.org/wiki/Goldbach%27s_conjecture Goldbach's Conjecture] & [http://en.wikipedia.org/wiki/Twin_prime 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 [http://www.slideshare.net/ChrisDeCorte1/finding-goldbach-q 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 [http://en.wikipedia.org/wiki/Prime-counting_function 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 [http://www.slideshare.net/ChrisDeCorte1/probabilistic-approach-to-prime-counting 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 [http://www.youtube.com/watch?v=Un1z3iJXVtA 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 [http://www.slideshare.net/ChrisDeCorte1/better-prime-counting-formula seems] to be better than the pure version of the [http://en.wikipedia.org/wiki/Logarithmic_integral 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 [http://www.slideshare.net/ChrisDeCorte1/approximating-the-trisection-of-an-angle approximation] to the [http://en.wikipedia.org/wiki/Angle_trisection angle trisection problem]

He also found an [http://www.slideshare.net/ChrisDeCorte1/approximations-in-drawing-and-squaring-the-circle approximation] to the [http://en.wikipedia.org/wiki/Squaring_the_circle 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 [http://www.slideshare.net/ChrisDeCorte1/fractal-approximations-tosomefamousconstants approximation] of some famous constants as a fractal.