Data update

Task/Pells-equation/Ada/pells-equation.adb

@@ -0,0 +1,65 @@
+with Ada.Text_Io;
+with Ada.Numerics.Elementary_Functions;
+with Ada.Numerics.Big_Numbers.Big_Integers;
+
+procedure Pells_Equation is
+   use Ada.Numerics.Big_Numbers.Big_Integers;
+
+   type Pair is
+      record
+         V1, V2 : Big_Integer;
+      end record;
+
+   procedure Solve_Pell (N : Natural; X, Y : out Big_Integer) is
+      use Ada.Numerics.Elementary_Functions;
+      Big_N : constant Big_Integer := To_Big_Integer (N);
+      XX    : constant Big_Integer := To_Big_Integer (Natural (Float'Floor (Sqrt (Float (N)))));
+   begin
+      if XX**2 = Big_N then
+         X := 1; Y := 0;
+         return;
+      end if;
+
+      declare
+         YY   : Big_Integer := XX;
+         Z    : Big_Integer := 1;
+         R    : Big_Integer := 2 * XX;
+         E    : Pair        := Pair'(V1 => 1, V2 => 0);
+         F    : Pair        := Pair'(V1 => 0, V2 => 1);
+      begin
+         loop
+            YY := R * Z - YY;
+            Z  := (Big_N - YY**2) / Z;
+            R  := (XX + YY) / Z;
+            E  := Pair'(V1 => E.V2, V2 => R * E.V2 + E.V1);
+            F  := Pair'(V1 => F.V2, V2 => R * F.V2 + F.V1);
+            X  := E.V2 + XX * F.V2;
+            Y  := F.V2;
+            exit when X**2 - Big_N * Y**2 = 1;
+         end loop;
+      end;
+   end Solve_Pell;
+
+   procedure Test (N : Natural) is
+
+      package Natural_Io is new Ada.Text_Io.Integer_Io (Natural);
+      use Ada.Text_Io, Natural_Io;
+
+      X, Y : Big_Integer;
+   begin
+      Solve_Pell (N, X => X, Y => Y);
+      Put ("X**2 - ");
+      Put (N, Width => 3);
+      Put (" * Y**2 = 1 for X = ");
+      Put (To_String (X, Width => 22));
+      Put (" and Y = ");
+      Put (To_String (Y, Width => 20));
+      New_Line;
+   end Test;
+
+begin
+   Test (61);
+   Test (109);
+   Test (181);
+   Test (277);
+end Pells_Equation;

Task/Pells-equation/Fortran/pells-equation.f

@@ -0,0 +1,53 @@
+      PROGRAM Pell
+      IMPLICIT NONE
+      INTEGER n(4)
+      DATA n /61, 109, 181, 277/
+      INTEGER*16 x, y
+      INTEGER i
+
+      DO 10, i = 1, size(n)
+         CALL solve(n(i), x, y)
+10       PRINT 100, n(i), x, y
+
+100   FORMAT ('x² - ', i3, 'y² = 1; x = ', i24, ' and y = ', i24)
+      END PROGRAM
+
+
+      SUBROUTINE solve(n, a, b)
+      IMPLICIT INTEGER*16 (A-Z)
+      INTEGER n
+
+      x = INT(sqrt(REAL(n)))
+      IF (x**2 .ne. n) GOTO 10
+
+      a = 1 ! perfect square; only trivial solution exists
+      b = 0
+      RETURN
+
+10    y = x
+      z = 1
+      r = 2*x
+      e1 = 1
+      e2 = 0
+      f1 = 0
+      f2 = 1
+
+20    y = r*z - y
+      z = (n - y**2) / z
+      r = (x + y) / z
+
+      e3 = r*e2 + e1
+      e1 = e2
+      e2 = e3
+
+      f3 = r*f2 + f1
+      f1 = f2
+      f2 = f3
+
+      a = e2 + x*f2
+      b = f2
+
+      IF (a**2 - n*b**2 .eq. 1) RETURN
+      GOTO 20
+
+      END SUBROUTINE

Task/Pells-equation/Langur/pells-equation.langur

@@ -1,4 +1,4 @@
-val fun = fn a, b, c: [b, b * c + a]
+val fun = fn(a, b, c) { [b, b * c + a] }
 
 val solvePell = fn(n) {
     val x = trunc(n ^/ 2)
@@ -17,7 +17,7 @@ val solvePell = fn(n) {
 }
 
 val C = fn(x) {
-    # format number string with commas
+    # format integer string with commas
     var neg, s = "", x -> string
     if s[1] == '-' {
         neg, s = "-", less(s, of=1)

Task/Pells-equation/Pluto/pells-equation.pluto

@@ -0,0 +1,29 @@
+require "bignum"
+local fmt = require "fmt"
+
+local function solve_pell(n)
+    n = bigint.new(n)
+    local x = bigint.sqrt(n)
+    local y = x
+    local z = bigint.one
+    local r = x * 2
+    local e1 = bigint.one
+    local e2 = bigint.zero
+    local f1 = bigint.zero
+    local f2 = bigint.one
+    while true do
+        y = r * z - y
+        z = (n - y * y) / z
+        r = (x + y) / z
+        e1, e2 = e2, r * e2 + e1
+        f1, f2 = f2, r * f2 + f1
+        local a = e2 + x * f2
+        local b = f2
+        if (a * a - n * b * b == bigint.one) then return {a, b} end
+    end
+end
+
+for {61, 109, 181, 277} as n do
+    local [a, b] = solve_pell(n)
+    fmt.print("x² - %3dy² = 1 for x = %-21s and y = %s", n, a, b)
+end