The ABC conjecture [La conjecture ABC].




The horizontal and vertical axes display respectively the whole numbers from 1 to N. Each disk display a couple of coprime numbers A (X axis) and B (Y axis):
                    GCD(A,B)=1
The number C is the sum of A and B:
                    C = A+B
The function Radical(N) gives the product of the prime factors (with an exponent equals to 1) of N. For example:
                                3  1  2
                    N = 1960 = 2 .5 .7
                                     1  1  1
                    Radical(1960) = 2 .5 .7  = 2.5.7 = 70


Then the following function is computed:
                                      log(C)
                    k(A,B,C) = ---------------------
                                log(Radical(A.B.C))
The ABC conjecture states that k(A,B,C) is less than a certain constant (unknown, but greater than 1 and hopefully lesser than 2...) whatever the values of A and B.



The surface and the luminance of each disk are proportional to k(A,B,C).

For this picture, the numbers A and B belong to [1,N=20] giving birth to the following values:
                    min(k(A,B,C))=0.38418327328527
                    max(k(A,B,C))=1.22629438553090



See some tridimensional visualizations:




See some related pictures (including this one):




(CMAP28 WWW site: this page was created on 09/26/2012 and last updated on 10/05/2015 15:14:17 -CEST-)



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