Goldbach Conjecture - DrD - 03-15-2015 11:45 AM
Here's a short program to display both the number of Goldbach partitions, and a list containing them:
For additional "fun" points, can you graph the number of resulting partitions vs. an even number (greater than 4) that produces a composition of twin prime summands?
Code:
//============================================
// Goldbach Conjecture, June 7,1742 =
// =
// Lists the number of Goldbach twin-prime =
// summand pairs, =
// Enter an even number greater than 4 =
// Displays Number of, and List of =
// Goldbach Partitions. =
// =
// ===========================================
EXPORT Gold(n)
BEGIN
local a:=0, i ;
L1:={};
print();
for i from 3 to n do
if isprime(i) and isprime(n-i) then
L1:=concat(L1,{{i,(n-i)}}); // Full Goldbach Composition
a:= a+1 ; // Total number of Goldbach elements
end;
end;
print("Goldbach Partitions " + a/2); // Number of distinct Goldbach partitions
print(L1({1,size(L1)/2})); // Goldbach partition listing
END;
RE: Goldbach Conjecture - DrD - 03-22-2015 05:32 AM
The code below is an attempt to plot the Goldbach "Comet," which is a plot of the number of twin prime summands, for each positive even integer, (up to a users specified number, and within calculator limitations). The plotted cometary tail gets more interesting for large numbers, (like 500), taking a correspondingly longer time to present, naturally.
I'm sure there are better ways to produce this plot, but readers may find interesting tidbits in the programming details useful for your own future projects, which is my primary purpose in sharing this program. I found the subject matter interesting, and fun, getting built to this point. Perhaps you will, too.
Seems to run ok on r6975 release firmware. If it locks up the calc, try: On + Symb, and re-run the program. The virtual calc does better.
-Dale-
Code:
//====================================================
// =
// Gold(n) =
// =
// Input: Positive EVEN integer greater than 2 =
// Output: Goldbach's Comet plot =
// =
// Descr: The composite number of twin prime summands =
// is plotted vs the positive even number >2 input. =
// =
//====================================================
// Goldbach conjecture (June 7, 1742): Positive even integers greater than 2 are the sum ot two prime numbers.
Goldbach();
rescale();
EXPORT Gold(n)
BEGIN
local a:=2,b,c,f,i,x,y;
local white:=rgb(255,255,255),
black:=rgb(0,0,0),
red:=rgb(255,0,0),
blue:=rgb(0,0,255),
yellow:=rgb(255,255,0);
Goldbach(n); // Determine vertical scale Max G(E) twin prime summands
c:=3*size(L1);
// Display Title: Goldbach's Comet
rect(0,20,320,214);
rect_p(75,1,265,16,blue);
textout_p("Goldbach's Comet",G0,110,4,2,yellow);
// Display plot screen
rect_p(40,20,300,200,white,black); // Plot field rectangle
// Vertical Axis G(E) Goldbach partitions
for a from 20 to 200 step 6 do
IFTE((a mod 5) ==0, line_p(40,a,45,a,yellow), line_p(40,a,42,a,yellow));
If (a mod 5) ==0 then
HDigits:=0;
b:=c-rescale(20,200,0,c,a);
textout_p(b,G0,IFTE((b<10),32,35-5*trunc((1+LOG(b)))),a,1,black);
HDigits:=2;
end; // end if
end; // Ends for: vert scale
// Horiz Axis Goldbach number E
for a from 40 to 300 step 26 do
IFTE((a mod 5) ==0, line_p(a,200,a,195,yellow), line_p(a,200,a,197,yellow));
If (a mod 5) ==0 then
HDigits:=0;
b:=rescale(40,300,0,n,a);
textout_p(b,G0,a,201,1,black);
HDigits:=2;
end; // end if
end; // Ends for: Horiz scale
// Composite G(E) for all positive even integers greater than 2 up to n
f:=4;
WHILE f <= n DO
Goldbach(f); // Get L1 (list of twin primes for each positive even integer greater than 2, up to n).
// Plot Goldbach composite data from L1
if size(L1) <> 0 then
for i from 1 to size(L1) do
x:=rescale(0,n,40,300,f);
y:=rescale(0,c,200,20,size(L1));
Case
if ip(y mod 4) == 0 then pixon_P(G0,x,y,white);end;
if ip(y mod 4) == 1 then pixon_P(G0,x,y,yellow);end;
if ip(y mod 4) == 2 then pixon_P(G0,x,y,red);end;
pixon_P(G0,x,y,blue);
end; // end case
end; // End for: plot
end; //end if L1 !={}
f:=f+2;
END; // ends while
wait(65535);
END; // Ends program
// ========================== Subroutines ==================================
//======================================
// Goldbach(n) =
// =
// Input: Current Positive Even Integer > 2 =
// Output: List of twin prime summands =
// =
//=====================================
Goldbach(n)
BEGIN
local a:=0,i;
// Make list of twin-prime number pairs
L1:={};
for i from 3 to n do
if isprime(i) and isprime(n-i) then
L1:=concat(L1,{{i,(n-i)}}); // Full Goldbach Composition
a:= a+1 ; // Total number of Goldbach elements
end; // end if
end; // end for
END;
// ============================================
// =
// Rescale =
// =
// Input: Datapoint d in range (old min, old max) =
// Returns: Value in range (new min, new max) =
// =
// ============================================
rescale(omn,omx,nmn,nmx,d)
BEGIN
RETURN (((d-omn)*(nmx-nmn))/(omx-omn))+nmn;
END;
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