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

4 = 2+2 6 = 3+3 8 = 3+5 [= 5+3] 10 = 3+7 = 5+5 [= 7+3] (...) 70 = 3+67 = 11+59 = 17+53 = 23+47 = 29+41 [= 41+29 = 47+23 = 53+17 = 59+11 = 67+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+P2In 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+PThis 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+2hence:

(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 09/30/2020 10:28:57 -CEST-)

[See all related pictures (including this one) [

[Please visit the related NumberTheory picture gallery [

[Go back to AVirtualSpaceTimeTravelMachine [

[

[

[Mail [

[About Pictures and Animations [

Copyright (c) CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / Ecole Polytechnique, 2012-2020.