Firstly, I have obtained an equation for the Golden Ratio. This is indeed a big feat. This gives the golden ratio value accurately (unlike my previous blog post on golden ratio with agm logic).
Well, the equation that gives the golden ratio phi is the solution for n in
(((n+1)^2)/((n+2)^2)-((n-1)^2)/(n^2))/(((n+1)/(n+2)-(n-1)/n)^2)=0. This is from the rational number series. Actually the equation can be simplified to n^2+n-1=0.
Thus the golden ratio value as -1.6180339887. . . . . . . . . is obtained.
Secondly, a prime number generator as effective as the Euler prime number generating polynomial is now made from the Rational number series.
The equation is
(((n+1)^2)/((n+2)^2)-((n-1)^2)/(n^2))/(((n+1)/(n+2)-(n-1)/n)^2)+42 generates prime numbers.
I have tested for 1 to 400. As much as 67% (268) of the results are prime.
Basically Euler prime generating formula was k^2-k+41. Mine is equivalent to k^2+k+41.
But both the above results can be obtained from the Rational number series. i.e the series
(((n+1)^2)/((n+2)^2)-((n-1)^2)/(n^2))/(((n+1)/(n+2)-(n-1)/n)^2).(equivalent to n^2+n-1)
So, the results are original.I feel privileged and feel honored to have got two of the finest results of mathematics. The Golden Ratio. And the Euler Prime Generating Polynomial.
Have great day!! Hope we all get freedom from the Covid-19 pandemic soon !!
Well, the equation that gives the golden ratio phi is the solution for n in
(((n+1)^2)/((n+2)^2)-((n-1)^2)/(n^2))/(((n+1)/(n+2)-(n-1)/n)^2)=0. This is from the rational number series. Actually the equation can be simplified to n^2+n-1=0.
Thus the golden ratio value as -1.6180339887. . . . . . . . . is obtained.
Secondly, a prime number generator as effective as the Euler prime number generating polynomial is now made from the Rational number series.
The equation is
(((n+1)^2)/((n+2)^2)-((n-1)^2)/(n^2))/(((n+1)/(n+2)-(n-1)/n)^2)+42 generates prime numbers.
I have tested for 1 to 400. As much as 67% (268) of the results are prime.
Basically Euler prime generating formula was k^2-k+41. Mine is equivalent to k^2+k+41.
But both the above results can be obtained from the Rational number series. i.e the series
(((n+1)^2)/((n+2)^2)-((n-1)^2)/(n^2))/(((n+1)/(n+2)-(n-1)/n)^2).(equivalent to n^2+n-1)
So, the results are original.I feel privileged and feel honored to have got two of the finest results of mathematics. The Golden Ratio. And the Euler Prime Generating Polynomial.
Have great day!! Hope we all get freedom from the Covid-19 pandemic soon !!
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