Goldbach's comet: Difference between revisions
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G(1000000) = 5402 |
G(1000000) = 5402 |
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Here, you can find the result of the visualization - or the "Goldbach's comet": [https://i.ibb.co/JvLDRc0/Screenshot-2022-05-07-at-11-35-23.png 2D-chart of all G values up to 2000] |
Here, you can find the result of the visualization - or the "Goldbach's comet": [https://i.ibb.co/JvLDRc0/Screenshot-2022-05-07-at-11-35-23.png 2D-chart of all G values up to 2000] |
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=={{header|Julia}== |
=={{header|Julia}== |
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Revision as of 17:29, 7 May 2022
You are encouraged to solve this task according to the task description, using any language you may know.
| This page uses content from Wikipedia. The original article was at Goldbach's comet. The list of authors can be seen in the page history. As with Rosetta Code, the text of Wikipedia is available under the GNU FDL. (See links for details on variance) |
Goldbach's comet is the name given to a plot of the function g(E), the so-called Goldbach function.
The Goldbach function is studied in relation to Goldbach's conjecture. The function g(E) is defined for all even integers E>2 to be the number of different ways in which E can be expressed as the sum of two primes.
- Examples
- G(4) = 1, since 4 can only be expressed as the sum of one distinct pair of primes (4 = 2 + 2)
- G(22) = 3, since 22 can be expressed as the sum of 3 distinct pairs of primes (22 = 11 + 11 = 5 + 17 = 3 + 19)
- Task
- Find and show (preferably, in a neat 10x10 table) the first 100 G numbers (that is: the result of the G function described above, for the first 100 even numbers >= 4)
- Find and display the value of G(1000000)
- Stretch
- Calculate the values of G up to 2000 (inclusive) and display the results in a scatter 2d-chart, aka the Goldbach's Comet
- See also
Arturo
<lang rebol>G: function [n][
size select 2..n/2 'x ->
and? [prime? x][prime? n-x]
]
print "The first 100 G values:" loop split.every: 10 map select 4..202 => even? => G 'row [
print map to [:string] row 'item -> pad item 3
]
print ["\nG(1000000) =" G 1000000]
csv: join.with:",\n" map select 4..2000 => even? 'x ->
~"|x|, |G x|"
- write the CSV data to a file which we can then visualize
- via our preferred spreadsheet app
write "comet.csv" csv</lang>
- Output:
The first 100 G values: 1 1 1 2 1 2 2 2 2 3 3 3 2 3 2 4 4 2 3 4 3 4 5 4 3 5 3 4 6 3 5 6 2 5 6 5 5 7 4 5 8 5 4 9 4 5 7 3 6 8 5 6 8 6 7 10 6 6 12 4 5 10 3 7 9 6 5 8 7 8 11 6 5 12 4 8 11 5 8 10 5 6 13 9 6 11 7 7 14 6 8 13 5 8 11 7 9 13 8 9 G(1000000) = 5402 Here, you can find the result of the visualization - or the "Goldbach's comet": [https://i.ibb.co/JvLDRc0/Screenshot-2022-05-07-at-11-35-23.png 2D-chart of all G values up to 2000]
{{header|Julia}
Run in VS Code or REPL to view and save the plot. <lang ruby>using Combinatorics using Plots using Primes
g(n) = iseven(n) ? count(p -> all(isprime, p), partitions(n, 2)) : error("n must be even")
println("The first 100 G numbers are: ")
foreach(p -> print(lpad(p[2], 4), p[1] % 10 == 0 ? "\n" : ""), map(g, 4:2:202) |> enumerate)
println("\nThe value of G(1000000) is ", g(1_000_000))
x = collect(4:2:4002) y = map(g, x) scatter(x .÷ 2, y, color = ["red", "blue", "green"][(y .÷ 2) .% 3 .+ 1])
</lang>
The first 100 G numbers are: 1 1 1 2 1 2 2 2 2 3 3 3 2 3 2 4 4 2 3 4 3 4 5 4 3 5 3 4 6 3 5 6 2 5 6 5 5 7 4 5 8 5 4 9 4 5 7 3 6 8 5 6 8 6 7 10 6 6 12 4 5 10 3 7 9 6 5 8 7 8 11 6 5 12 4 8 11 5 8 10 5 6 13 9 6 11 7 7 14 6 8 13 5 8 11 7 9 13 8 9 The value of G(1000000) is 5402
Raku
For the stretch, actually generates a plot, doesn't just calculate values to be plotted by a third party application. Deviates slightly from the stretch goal, plots the first two thousand defined values rather than the values up to two thousand that happen to be defined. (More closely matches the wikipedia example image.)
<lang perl6>sub G (Int $n) { +(2..$n/2).grep: { .is-prime && ($n - $_).is-prime } }
- Task
put "The first 100 G values:\n", (^100).map({ G 2 × $_ + 4 }).batch(10)».fmt("%2d").join: "\n";
put "\nG 1_000_000 = ", G 1_000_000;
- Stretch
use SVG; use SVG::Plot;
my @x = map 2 × * + 4, ^2000; my @y = @x.map: &G;
'Goldbachs-Comet-Raku.svg'.IO.spurt: SVG.serialize: SVG::Plot.new(
width => 1000, height => 500, background => 'white', title => "Goldbach's Comet", x => @x, values => [@y,],
).plot: :points; </lang>
- Output:
The first 100 G values: 1 1 1 2 1 2 2 2 2 3 3 3 2 3 2 4 4 2 3 4 3 4 5 4 3 5 3 4 6 3 5 6 2 5 6 5 5 7 4 5 8 5 4 9 4 5 7 3 6 8 5 6 8 6 7 10 6 6 12 4 5 10 3 7 9 6 5 8 7 8 11 6 5 12 4 8 11 5 8 10 5 6 13 9 6 11 7 7 14 6 8 13 5 8 11 7 9 13 8 9 G 1_000_000 = 5402
Stretch goal: (offsite SVG image) Goldbachs-Comet-Raku.svg
Wren
<lang ecmascript>import "./math" for Int import "./trait" for Stepped import "./fmt" for Fmt import "io" for File
var limit = 2000 var primes = Int.primeSieve(limit-1).skip(1).toList var goldbach = {4: 1} for (i in Stepped.new(6..limit, 2)) goldbach[i] = 0 for (i in 0...primes.count) {
for (j in i...primes.count) {
var s = primes[i] + primes[j]
if (s > limit) break
goldbach[s] = goldbach[s] + 1
}
}
System.print("The first 100 G values:") var count = 0 for (i in Stepped.new(4..202, 2)) {
count = count + 1
Fmt.write("$2d ", goldbach[i])
if (count % 10 == 0) System.print()
}
primes = Int.primeSieve(499999).skip(1) var gm = 0 for (p in primes) {
if (Int.isPrime(1e6 - p)) gm = gm + 1
} System.print("\nG(1000000) = %(gm)")
// create .csv file for values up to 2000 for display by an external plotter // the third field being the color (red = 0, blue = 1, green = 2) File.create("goldbachs_comet.csv") { |file|
for(i in Stepped.new(4..limit, 2)) {
file.writeBytes("%(i), %(goldbach[i]), %(i/2 % 3)\n")
}
}</lang>
- Output:
The first 100 G values: 1 1 1 2 1 2 2 2 2 3 3 3 2 3 2 4 4 2 3 4 3 4 5 4 3 5 3 4 6 3 5 6 2 5 6 5 5 7 4 5 8 5 4 9 4 5 7 3 6 8 5 6 8 6 7 10 6 6 12 4 5 10 3 7 9 6 5 8 7 8 11 6 5 12 4 8 11 5 8 10 5 6 13 9 6 11 7 7 14 6 8 13 5 8 11 7 9 13 8 9 G(1000000) = 5402
XPL0
<lang XPL0> func IsPrime(N); \Return 'true' if N is prime int N, I; [if N <= 2 then return N = 2; if (N&1) = 0 then \even >2\ return false; for I:= 3 to sqrt(N) do
[if rem(N/I) = 0 then return false; I:= I+1; ];
return true; ];
int PT(1_000_000);
func G(E); \Ways E can be expressed as sum of two primes int E, C, I, J, T; [C:= 0; I:= 0; loop [J:= I;
if PT(J) + PT(I) > E then return C;
loop [T:= PT(J) + PT(I);
if T = E then C:= C+1;
if T > E then quit;
J:= J+1;
];
I:= I+1;
];
];
int I, N; [I:= 0; \make prime table for N:= 2 to 1_000_000 do
if IsPrime(N) then
[PT(I):= N; I:= I+1];
I:= 4; \show first 100 G numbers Format(4, 0); for N:= 1 to 100 do
[RlOut(0, float(G(I)));
if rem(N/10) = 0 then CrLf(0);
I:= I+2;
];
CrLf(0); Text(0, "G(1,000,000) = "); IntOut(0, G(1_000_000)); CrLf(0); ]</lang>
- Output:
1 1 1 2 1 2 2 2 2 3 3 3 2 3 2 4 4 2 3 4 3 4 5 4 3 5 3 4 6 3 5 6 2 5 6 5 5 7 4 5 8 5 4 9 4 5 7 3 6 8 5 6 8 6 7 10 6 6 12 4 5 10 3 7 9 6 5 8 7 8 11 6 5 12 4 8 11 5 8 10 5 6 13 9 6 11 7 7 14 6 8 13 5 8 11 7 9 13 8 9 G(1,000,000) = 5402