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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Copyright (c) Jean-François Colonna, 2012-2015.

Copyright (c) CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / Ecole Polytechnique, 2012-2015.