Tridimenional display of the Goldbach conjecture [Visualisation tridimensionnelle de la conjecture de Goldbach].




The Goldbach conjecture states that each even integer number N greater or equal to 4 can be written as the sum of two prime numbers. For example: 
                    4   = 2+2
                    6   = 3+3
                    8   = 3+5 [= 5+3]
                    10  = 3+7 = 5+5 [= 7+3]
                    (...)
                    990 =   7+983 =  13+977 =  19+971 =  23+967 =  37+953 =  43+947 =  53+937 =  61+929 =  71+919 =  79+911
                        =  83+907 = 103+887 = 107+883 = 109+881 = 113+877 = 127+863 = 131+859 = 137+853 = 151+839 = 163+827
                        = 167+823 = 179+811 = 181+809 = 193+797 = 229+761 = 233+757 = 239+751 = 251+739 = 257+733 = 263+727
                        = 271+719 = 281+709 = 307+683 = 313+677 = 317+673 = 331+659 = 337+653 = 347+643 = 349+641 = 359+631
                        = 373+617 = 383+607 = 389+601 = 397+593 = 419+571 = 421+569 = 433+557 = 443+547 = 449+541 = 467+523
                        = 487+503 = 491+499
                       [= 499+491 = 503+487 = 523+467 = 541+449 = 547+443 = 557+433 = 569+421 = 571+419 = 593+397 = 601+389
                        = 607+383 = 617+373 = 631+359 = 641+349 = 643+347 = 653+337 = 659+331 = 673+317 = 677+313 = 683+307
                        = 709+281 = 719+271 = 727+263 = 733+257 = 739+251 = 751+239 = 757+233 = 761+229 = 797+193 = 809+181
                        = 811+179 = 823+167 = 827+163 = 839+151 = 853+137 = 859+131 = 863+127 = 877+113 = 881+109 = 883+107
                        = 887+103 = 907+83  = 911+79  = 919+71  = 929+61  = 937+53  = 947+43  = 953+37  = 967+23  = 971+19
                        = 977+13  = 983+7]
                    (...)
Each straight line (with equation y=-x+N) of this picture displays an even integer number N and each orange square with center coordinates {x=P1,y=P2} exhibits a couple of prime numbers P1 and P2 such that: 
                    N = P1+P2
In order to exhibit the commutativity of the addition, both {P1,P2} and {P2,P1} couples are displayed, hence the symmetry.

Here N varies between 2 and 33 (the first displayed white line is the N=6 line, when the N=4 line is the bottom left "corner" of the surface).


Let DS(N) be the Sum of all the Divisors of N. For example: 
                    SD(2) = 1+2 = 3
                    SD(3) = 1+3 = 4
                    SD(4) = 1+2+4 = 7
                    SD(5) = 1+5 = 6
                    SD(6) = 1+2+3+6 = 12
                    SD(7) = 1+7 = 8
                    (...)
When N is a Prime Number P: 
                    SD(P) = 1+P
This picture displays the surface {x,y,(SD(x)+SD(y))-(x+y+2)} and the luminance of each point is proportional to its third coordinate. For each even integer number N, the lowest points of the x+y=N straight line (white) correspond to couple {P1,P2} of Prime Numbers (orange squares). There is always at least one such orange square on each white line: this is the Goldbach conjecture. By the way: 
                    SD(P1)        = 1+P1
                    SD(P2)        = 1+P2
                    SD(P1)+SD(P2) = (1+P1)+(1+P2)
                    SD(P1)+SD(P2) = P1+P2+2
                    SD(P1)+SD(P2) = N+2
hence: 
                    (SD(P1)+SD(P2))-(N+2) = 0



See some related visualizations (including this one):

 
 
 
 



(CMAP28 WWW site: this page was created on 11/13/2012 and last updated on 10/20/2022 12:46:38 -CEST-)



[See all related pictures (including this one) [Voir toutes les images associées (incluant celle-ci)]]

[Please visit the related NumberTheory picture gallery [Visitez la galerie d'images NumberTheory associée]]

[Go back to AVirtualMachineForExploringSpaceTimeAndBeyond [Retour à AVirtualMachineForExploringSpaceTimeAndBeyond]]

[The Y2K Bug [Le bug de l'an 2000]]

[Site Map, Help and Search [Plan du Site, Aide et Recherche]]
[Mail [Courrier]]
[About Pictures and Animations [A Propos des Images et des Animations]]


Copyright © Jean-François Colonna, 2012-2022.
Copyright © CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / Ecole Polytechnique, 2012-2022.