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# Goldbach's comet

Goldbach's comet
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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)

• 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

## ALGOL 68

Generates an ASCII-Art scatter plot - the vertical axis is n/10 and the hotizontal is G(n).

`BEGIN # calculate values of the Goldbach function G where G(n) is the number #      # of prime pairs that sum to n, n even and > 2                         #      # generates an ASCII scatter plot of G(n) up to G(2000)                #      # (Goldbach's Comet)                                                   #    PR read "primes.incl.a68" PR          # include prime utilities          #    INT max prime = 1 000 000;            # maximum number we will consider  #    INT max plot  =     2 000;            # maximum G value for the comet    #    []BOOL prime = PRIMESIEVE max prime;  # sieve of primes to max prime     #    [ 0 : max plot ]INT g2;               # table of G values: g2[n] = G(2n) #    # construct the table of G values #    FOR n FROM LWB g2 TO UPB g2 DO g2[ n ] := 0 OD;    g2[ 4 ] := 1;                     # 4 is the only sum of two even primes #    FOR p FROM 3 BY 2 TO max plot OVER 2 DO        IF prime[ p ] THEN            g2[ p + p ] +:= 1;            FOR q FROM p + 2 BY 2 TO max plot - p DO                IF prime[ q ] THEN                    g2[ p + q ] +:= 1                FI            OD        FI    OD;    # show the first hundred G values #    INT c := 0;    FOR n FROM 4 BY 2 TO 202 DO        print( ( whole( g2[ n ], -4 ) ) );        IF ( c +:= 1 ) = 10 THEN print( ( newline ) ); c := 0 FI    OD;    # show G( 1 000 000 ) #    INT gm := 0;    FOR p FROM 3 TO max prime OVER 2 DO        IF prime[ p ] THEN            IF prime[ max prime - p ] THEN                gm +:= 1            FI        FI    OD;    print( ( "G(", whole( max prime, 0 ), "): ", whole( gm, 0 ), newline ) );    # find the maximum value of G up to the maximum plot size #    INT max g := 0;    FOR n FROM 2 BY 2 TO max plot DO        IF g2[ n ] > max g THEN max g := g2[ n ] FI    OD;    # draw an ASCII scatter plot of G, each position represents 5 G values #    # the vertical axis is n/10, the horizontal axis is G(n) #    INT plot step = 10;    STRING plot value = " .-+=*%\$&#@";    FOR g FROM 0 BY plot step TO max plot - plot step DO        [ 0 : max g ]INT values;        FOR v pos FROM LWB values TO UPB values DO values[ v pos ] := 0 OD;        INT max v := 0;        FOR g element FROM g BY 2 TO g + ( plot step - 1 ) DO            INT g2 value = g2[ g element ];            values[ g2 value ] +:= 1;            IF g2 value > max v THEN max v := g2 value FI        OD;        print( ( IF g MOD 100 = 90 THEN "+" ELSE "|" FI ) );        FOR v pos FROM 1 TO max v DO # exclude 0 values from the plot #            print( ( plot value[ values[ v pos ] + 1 ] ) )        OD;        print( ( newline ) )    ODEND`
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(1000000): 5402
```
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```

## Arturo

`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 appwrite "comet.csv" csv`
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": 2D-chart of all G values up to 2000

## AWK

` # syntax: GAWK -f GOLDBACHS_COMET.AWKBEGIN {    print("The first 100 G numbers:")    for (n=4; n<=202; n+=2) {      printf("%4d%1s",g(n),++count%10?"":"\n")    }    n = 1000000    printf("\nG(%d): %d\n",n,g(n))    n = 4    printf("G(%d): %d\n",n,g(n))    n = 22    printf("G(%d): %d\n",n,g(n))    exit(0)}function g(n,  count,i) {    if (n % 2 == 0) { # n must be even      for (i=2; i<=(1/2)*n; i++) {        if (is_prime(i) && is_prime(n-i)) {          count++        }      }    }    return(count)}function is_prime(n,  d) {    d = 5    if (n < 2) { return(0) }    if (n % 2 == 0) { return(n == 2) }    if (n % 3 == 0) { return(n == 3) }    while (d*d <= n) {      if (n % d == 0) { return(0) }      d += 2      if (n % d == 0) { return(0) }      d += 4    }    return(1)} `
Output:
```The first 100 G numbers:
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
G(4): 1
G(22): 3
```

## FreeBASIC

`Function isPrime(Byval ValorEval As Uinteger) As Boolean    If ValorEval <= 1 Then Return False    For i As Integer = 2 To Int(Sqr(ValorEval))        If ValorEval Mod i = 0 Then Return False    Next i    Return TrueEnd Function Function g(n As Uinteger) As Uinteger    Dim As Uinteger i, count = 0    If (n Mod 2 = 0) Then     'n in goldbach function g(n) must be even        For i = 2 To (1/2) * n            If isPrime(i) And isPrime(n - i) Then count += 1        Next i    End If    Return countEnd Function Print "The first 100 G numbers are:" Dim As Uinteger col = 1For n As Uinteger = 4 To 202 Step 2    Print Using "####"; g(n);    If (col Mod 10 = 0) Then Print    col += 1Next n Print !"\nThe value of G(1000000) is "; g(1000000)Sleep`
Output:
```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```

## J

`   10 10\$#/.~4,/:~ 0-.~,(<:/~ * +/~) p:1+i.p:inv 202 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  811  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`

And, for G(1e6):

`    -:+/1 p: 1e6-p:i.p:inv 1e65402`

Explanation:

For the first part, the easiest approach seems to be to brute force it. The first 100 numbers starting with 4 end at 202, so we start by finding all primes less than 202 (and sum all pairs of these primes).

Also, we can eliminate an odd/even test on these sums of primes by excluding 2 from our list of primes and including 4 as an explicit constant.

Also, since we are only concerned about distinct sums, we eliminate all pairs where the first prime in the sum exceeds the second prime in the sum.

So.. we compute a couple thousand sums, sort them in numeric order (prepending that list with 4), count how many times each unique value appears, and order the first 100 of those frequencies in a 10x10 table.

---

For G(1e6) that brute force approach becomes inefficient -- instead of about 2000 sums, almost 600 of which are relevant, we would need to find over 6e9 sums, and about 6e9 of them would be irrelevant.

Instead, for G(1e6), we find all primes less than a million, subtract each from 1 million and count how many of the differences are prime and cut that in half. We cut that sum in half because this approach counts each pair twice (once with the smallest value first, again with the smallest value second -- since 1e6 is not the square of a prime we do not have a prime which appears twice in one of these sums).

## Julia

Run in VS Code or REPL to view and save the plot.

`using Combinatoricsusing Plotsusing 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(2:2002)y = map(g, 2x)scatter(x, y, markerstrokewidth = 0, color = ["red", "blue", "green"][mod1.(x, 3)]) `
Output:
```
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

```

## Perl

Library: ntheory
`use strict;use warnings;use feature 'say'; use List::Util 'max';use GD::Graph::bars;use ntheory 'is_prime'; sub table { my \$t = shift() * (my \$c = 1 + max map {length} @_); ( sprintf( ('%'.\$c.'s')x@_, @_) ) =~ s/.{1,\$t}\K/\n/gr } sub G {    my(\$n) = @_;    scalar grep { is_prime(\$_) and is_prime(\$n - \$_) } 2 .. \$n/2;} my @y;push @y, G(2*\$_ + 4) for my @x = 0..1999; say \$_ for table 10, @y;printf "G \$_: %d", G(\$_) for 1e6; my @data = ( \@x, \@y);my \$graph = GD::Graph::bars->new(1200, 400);\$graph->set(    title          => q/Goldbach's Comet/,    y_max_value    => 170,    x_tick_number  => 10,    r_margin       => 10,    dclrs          => [ 'blue' ],) or die \$graph->error;my \$gd = \$graph->plot(\@data) or die \$graph->error; open my \$fh, '>', 'goldbachs-comet.png';binmode \$fh;print \$fh \$gd->png();close \$fh;`
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 1000000: 5402```

Stretch goal: (offsite image) goldbachs-comet.png

## Phix

Library: Phix/pGUI
Library: Phix/online

You can run this online here.

```--
-- demo\rosetta\Goldbachs_comet.exw
-- ================================
--
-- Note: this plots n/2 vs G(n) for n=6 to 4000 by 2, matching wp and
--       Algol 68, Python, and Raku. However, while not wrong, Arturo
--       and Wren apparently plot n vs G(n) for n=6 to 2000 by 2, so
--       should you spot any (very) minor differences, that'd be why.
--
with javascript_semantics
requires("1.0.2")

constant limit = 4000
sequence primes = get_primes_le(limit)[2..\$],
goldbach = reinstate(repeat(0,limit),{4},{1})
for i=1 to length(primes) do
for j=i to length(primes) do
integer s = primes[i] + primes[j]
if s>limit then exit end if
goldbach[s] += 1
end for
end for

sequence fhg = extract(goldbach,tagstart(4,100,2))
string fhgs = join_by(fhg,1,10," ","\n","%2d")
integer gm = sum(apply(sq_sub(1e6,get_primes_le(499999)[2..\$]),is_prime))
printf(1,"The first 100 G values:\n%s\n\nG(1,000,000) = %,d\n",{fhgs,gm})

include pGUI.e
include IupGraph.e

function get_data(Ihandle /*graph*/)
return {{tagset(limit/2,3,3),extract(goldbach,tagset(limit,6,6)),CD_RED},
{tagset(limit/2,4,3),extract(goldbach,tagset(limit,8,6)),CD_BLUE},
{tagset(limit/2,5,3),extract(goldbach,tagset(limit,10,6)),CD_DARK_GREEN}}
end function

IupOpen()
Ihandle graph = IupGraph(get_data,"RASTERSIZE=640x440,MARKSTYLE=PLUS")
IupSetAttributes(graph,"XTICK=%d,XMIN=0,XMAX=%d",{limit/20,limit/2})
IupSetAttributes(graph,"YTICK=20,YMIN=0,YMAX=%d",{max(goldbach)+20})
Ihandle dlg = IupDialog(graph,`TITLE="Goldbach's comet",MINSIZE=400x300`)
IupShow(dlg)
if platform()!=JS then
IupMainLoop()
IupClose()
end if
```
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) = 5,402
```

## Python

`from matplotlib.pyplot import scatter, showfrom sympy import isprime def g(n):    assert n > 2 and n % 2 == 0, 'n in goldbach function g(n) must be even'              count = 0    for i in range(1, n//2 + 1):        if isprime(i) and isprime(n - i):            count += 1    return count print('The first 100 G numbers are:') col = 1for n in range(4, 204, 2):    print(str(g(n)).ljust(4), end = '\n' if (col % 10 == 0) else '')    col += 1 print('\nThe value of G(1000000) is', g(1_000_000)) x = range(4, 4002, 2)y = [g(i) for i in x]colors = [["red", "blue", "green"][(i // 2) % 3] for i in x]scatter([i // 2 for i in x], y, marker='.', color = colors)show()`
Output:
```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.)

`sub G (Int \$n) { +(2..\$n/2).grep: { .is-prime && (\$n - \$_).is-prime } } # Taskput "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; # Stretchuse 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; `
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

Library: Wren-math
Library: Wren-trait
Library: Wren-fmt
`import "./math" for Intimport "./trait" for Steppedimport "./fmt" for Fmtimport "io" for File var limit = 2000var primes = Int.primeSieve(limit-1).skip(1).toListvar goldbach = {4: 1}for (i in Stepped.new(6..limit, 2)) goldbach[i] = 0for (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 = 0for (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 = 0for (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")    }}`
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

` func IsPrime(N); \Return 'true' if N is primeint  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 primesint  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 tablefor N:= 2 to 1_000_000 do        if IsPrime(N) then                [PT(I):= N;  I:= I+1];I:= 4;  \show first 100 G numbersFormat(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);]`
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
```