НАСТОЯЩАЯ ИСТОРИЯ БУДУЩЕГО

CHF PSEPHOS CHF

% Тёк чернобелый песок, тёк.

% Черный тёк не низок и белый тёк не высок.

REFERENCES

% The Ancient Greek ψηφ method of counting with black 阴 черный and white 阳 белый stones.
% ψῆφος, psephos is a Greek word meaning pebble, галька.

% Jeffrey C. Lagarias & Harsh Mehta, Products of Farey Fractions, Experimental Mathematics, Volume 26, 2017 - Issue 1.

% Franel, J.; Landau, E. (1924), "Les suites de Farey et le problème des nombres premiers" (Franel, 198-201);
% "Bemerkungen zu der vorstehenden Abhandlung von Herrn Franel (Landau, 202-206)", Göttinger Nachrichten: 198-206.

% By Mr. J. Farey, Sen. To Mr. Tilloch. "On a curious Property of vulgar Fractions."
% Philosophical Magazine (1816), vol. 17 nr. 217, blz. 385-386.

% One stone     Ein Stein     Eén steen     Une pierre     Unus lapis     Μια πέτρα     Один камень
% Two stones    Zwei Steine   Twee stenen   Deux pierres   Duo lapides    Δύο πέτρες    Два камня

% Chf means cijfer, cypher, ψιφρα, цифра.
% Psephos generalize Farey fractions.

Triangle UF begins

%             -1                                                   0;
%          -2 -1                     1      -1                     0 0;
%       -3 -2 -1           2 -2      1      -1      1 -1           0 0 0;
%    -4 -3 -2 -1      3 -3 2 -2      1 2 -2 -1      1 -1 1 -1      0 0 0 0;
% -5 -4 -3 -2 -1 4 -4 3 -3 2 -2 3 -3 1 2 -2 -1 2 -2 1 -1 1 -1 1 -1 0 0 0 0 0;
                                      ....

Triangle OF begins

%             -1                                                   1;
%          -2 -1                     2      -2                     1 2;
%       -3 -2 -1           3 -3      2      -2      3 -3           1 2 3;
%    -4 -3 -2 -1      4 -4 3 -3      2 4 -4 -2      3 -3 4 -4      1 2 3 4;
% -5 -4 -3 -2 -1 5 -5 4 -4 3 -3 5 -5 2 4 -4 -2 5 -5 3 -3 4 -4 5 -5 1 2 3 4 5;
%                                      ....

Triangle VF begins

%              0                                                   1;
%           0  0                     1      -1                     1 2;
%        0  0  0           1 -1      1      -1      2 -2           1 2 3;
%     0  0  0  0      1 -1 1 -1      1 2 -2 -1      2 -2 3 -3      1 2 3 4;
%  0  0  0  0  0 1 -1 1 -1 1 -1 2 -2 1 2 -2 -1 3 -3 2 -2 3 -3 4 -4 1 2 3 4 5;
%                                      ....

INITIALISATION

clear all
clc

tot = 19;

UF = [-1  0];
OF = [-1 +1];
VF = [ 0 +1];

ITERATION

for n = 1:tot

 N(n) = n;

 normo = n*(n+1)-0;
 normi = n*(n+1)-1;

 %%%%%%%%%%%%%%%%%
 uf = UF; UF = [];
 of = OF; OF = [];
 vf = VF; VF = [];
 %%%%%%%%%%%%%%%%%

PSEPHOS CONSTRUCTION BY REGULAR INSERTION

 for i = 1:n
  UF = [UF [-n+i-1 uf((n-1)*(i-1)+(1:n-1)) n-i] ]; % PSEPHOS NOMINATORS <===
  OF = [OF [-n     of((n-1)*(i-1)+(1:n-1)) n  ] ]; % PSEPHOS DENOMINATORS ==  АНТИПАЛИНДРОМ
  VF = [VF [+0-i+1 vf((n-1)*(i-1)+(1:n-1)) i  ] ]; % PSEPHOS NOMINATORS ===>
 end

UNIFORMITY OF UNREDUCED FAREY CIRCUMCISION

-----------

 j = (0:1:n-1);

-----------

 UNIFORMITY_UF(n,:) = [all(UF(n+1:(n+1):n*(n+1)) == n-j-1) all(UF(0+1:(n+1):n*(n+1)) == -n+j)];
 UNIFORMITY_OF(n,:) = [all(OF(n+1:(n+1):n*(n+1)) == n    ) all(OF(0+1:(n+1):n*(n+1)) == -n  )];
 UNIFORMITY_VF(n,:) = [all(VF(n+1:(n+1):n*(n+1)) == j+1  ) all(VF(0+1:(n+1):n*(n+1)) == -j  )];

COMMON HOLISTIC FACTORS

 UOV = gcd(UF,VF);

ADJACENCY

 UF0 = UF; UF0(1) = []; UF1 = UF; UF1(end) = [];
 OF0 = OF; OF0(1) = []; OF1 = OF; OF1(end) = [];
 VF0 = VF; VF0(1) = []; VF1 = VF; VF1(end) = [];

 UV_VU(n) = sum(UF0.*VF1 - VF0.*UF1);

 SUMMI(n) = sum([VF.*normi-(0:normi).*OF]);

end

REGULAR FRACTIONS

II = (0:normi); regul = II./(normo); IO = II + 1;

PSEPHOS REDUCED FRACTIONS

pseph = VF./OF;

SUPERSYMMETRIC PSEPHOS

[II' UF' OF' VF'];

TOTIENT

TOTIENT = [1 diff(UV_VU)];

CONJECTURE TO BECOME A THEOREM

PSEPHOS_CONJECTURE = all(sum(abs(pseph-regul).^2) < 1./N );

UNIFORMITY

UNIFORMITY_OK = [all(UNIFORMITY_UF) all(UNIFORMITY_OF) all(UNIFORMITY_VF)];

CHF

for k = 1:normo;

 I_(k) = ns_(k);
 UF_(k) = ns_(UF(k));
 OF_(k) = ns_(OF(k));
 VF_(k) = ns_(VF(k));

 if 1 == 0
  zi = sign(OF(k));
  draw_binm_(k) = zcmsm_to_zbinm_pasen_2023_(+abs(UF(k)).*1,+abs(VF(k)).*1);
  draw_chfm_(k) = zcmsm_to_zchfm_pasen_2023_(+abs(UF(k))*zi,+abs(VF(k))*zi);
 end

 aUF = abs(UF(k));
 aOF = abs(OF(k));
 aVF = abs(VF(k));

CHRISTMATH FORMULA PASEN 2023

 cofacto(k) = (4*min(aUF,aVF)-aOF).*(gcd(aUF,aOF)  > 1);
 coprime(k) = (4*min(aUF,aVF)-aOF).*(gcd(aUF,aOF) == 1);
 formulo(k) = (4*min(aUF,aVF)-aOF);

 coprime(k) = (4*min(aUF,aVF)-aOF).*(gcd(aUF,aOF) == 1)*tot*(tot+1);
 cofacto(k) = (4*min(aUF,aVF)-aOF).*(gcd(aUF,aOF)  > 1)*tot*(tot+1)+tot+(mod(tot,2) == 1);
 formula(k) = (4*min(aUF,aVF)-aOF)                     *tot*(tot+1)+tot+(mod(tot,2) == 1);

end % k = 1:normo

FORMULA        = sum(formula);
FORMULAcoprime = sum(coprime);
FORMULAcofacto = sum(cofacto);

[FORMULA FORMULAcoprime FORMULAcofacto];

PSEPHOS UNREDUCED FAREY COPRIME & COFACTO EQUILIBRIUM

EQUAL_COPRIME_COFACTO_SUMS = all(coprime + cofacto == formula);

totient = TOTIENT(tot);
cotient = tot-totient;

PLOT

if 1 == 1

   figure(1); plot([ pseph' regul']);        title("PSEPHOS EQUIDISTRIBUTION");

   figure(2); plot([ pseph'-regul']);        title("PSEPHOS EQUIDISTRIBUTION");

   figure(3); plot(UF-VF,UF+VF,'b');         title("PSEPHOS EQUIDISTRIBUTION");

   figure(4); plot(UOV./OF,'.');             title("COMMON HOLISTIC FACTORS");

EQUAL COPRIME & COFACTO SUMS

   figure(5); hold on;
              plot(+coprime,IO,'r.');
              plot(-coprime,IO,'r.')
              plot(-cofacto,IO,'b.');
              plot(+cofacto,IO,'b.');        title("PASEN"); grid off; axis off;

%  figure(6); plot(cumsum(cofacto-coprime));
%  figure(7); plot(cumsum(coprime+cofacto));

%  figure(8); plot(OF,'w'); hold on;
%             plot(OF,'.'); grid on;         title("PSEPHOS EQUIDISTRIBUTION");  for i = 1:normo; text(i,OF(i),num2str(i)); end;

end

% ---------------------------------------------
% [I_' draw_chfm_' draw_binm_' UF_' OF_' VF_'];
% ---------------------------------------------

chf ψηφ

KNOWLEDGE TREASURY BY PSEPHOS ENUMERATION

ψηφ chf

% for The Current History of the Future
% НАСТОЯЩАЯ ИСТОРИЯ БУДУЩЕГО
%       ψηφ PSEPHOS chf
%    Цепная  форма  наследия
% ψηφ
% By I.V. Serov
% Delft & Троицк
% <03-06-2019 : 09-04-2023>
% ψηφ

Published with MATLAB® & chf.nu; 2023