From 098ecc99d417944ae521bbc643d06d8e3d9a9811 Mon Sep 17 00:00:00 2001
From: Max Horn <max@quendi.de>
Date: Mon, 15 Feb 2021 10:41:09 +0100
Subject: [PATCH 1/3] Add our own version of the Jacobi symbol

---
 gap/TestActionPackage.gi | 41 ++++++++++++++++++++++++++++++++++++++++
 1 file changed, 41 insertions(+)

diff --git a/gap/TestActionPackage.gi b/gap/TestActionPackage.gi
index c0a0a08..97369cf 100644
--- a/gap/TestActionPackage.gi
+++ b/gap/TestActionPackage.gi
@@ -8,3 +8,44 @@ function()
 	Print( "This is a placeholder function, replace it with your own code.\n" );
 end );
 
+BindGlobal("MyJacobi", function ( n, m )
+  local  jac, t;
+
+  # check the argument
+  if m <= 0 then
+    Error("<m> must be positive");
+  fi;
+
+  # compute the Jacobi symbol similar to Euclid's algorithm
+  jac := 1;
+  while m <> 1  do
+
+    # if the gcd of $n$ and $m$ is $>1$ Jacobi returns $0$
+    if n = 0 or (n mod 2 = 0 and m mod 2 = 0)  then
+      jac := 0;  m := 1;
+
+    # $J(n,2*m) = J(n,m) * J(n,2) = J(n,m) * (-1)^{(n^2-1)/8}$
+    elif m mod 2 = 0  then
+      if n mod 8 = 3  or  n mod 8 = 5  then jac := -jac;  fi;
+      m := m / 2;
+
+    # $J(2*n,m) = J(n,m) * J(2,m) = J(n,m) * (-1)^{(m^2-1)/8}$
+    elif n mod 2 = 0  then
+      if m mod 8 = 3  or  m mod 8 = 5  then jac := -jac;  fi;
+      n := n / 2;
+
+    # $J(-n,m) = J(n,m) * J(-1,m) = J(n,m) * (-1)^{(m-1)/2}$
+    elif n < 0  then
+      if m mod 4 = 3  then jac := -jac;  fi;
+      n := -n;
+
+    # $J(n,m) = J(m,n) * (-1)^{(n-1)*(m-1)/4}$ (quadratic reciprocity)
+    else
+      if n mod 4 = 3  and m mod 4 = 3  then jac := -jac;  fi;
+      t := n;  n := m mod n;  m := t;
+
+    fi;
+  od;
+
+  return jac;
+end);

From e915de866b3ab10f6f3aa61b84574ae4ab9f3576 Mon Sep 17 00:00:00 2001
From: Max Horn <max@quendi.de>
Date: Mon, 15 Feb 2021 11:54:12 +0100
Subject: [PATCH 2/3] Add missing test file

---
 tst/jacobi.tst | 3 +++
 1 file changed, 3 insertions(+)
 create mode 100644 tst/jacobi.tst

diff --git a/tst/jacobi.tst b/tst/jacobi.tst
new file mode 100644
index 0000000..446a001
--- /dev/null
+++ b/tst/jacobi.tst
@@ -0,0 +1,3 @@
+# Test MyJacobi
+gap> List([0..5], n -> MyJacobi(n, 5));
+[ 0, 1, -1, -1, 1, 0 ]

From c1de931ae2bbedd165bf9fbb53e561886ab86497 Mon Sep 17 00:00:00 2001
From: Max Horn <max@quendi.de>
Date: Mon, 15 Feb 2021 12:02:18 +0100
Subject: [PATCH 3/3] update

---
 tst/jacobi.tst | 2 ++
 1 file changed, 2 insertions(+)

diff --git a/tst/jacobi.tst b/tst/jacobi.tst
index 446a001..008274d 100644
--- a/tst/jacobi.tst
+++ b/tst/jacobi.tst
@@ -1,3 +1,5 @@
 # Test MyJacobi
 gap> List([0..5], n -> MyJacobi(n, 5));
 [ 0, 1, -1, -1, 1, 0 ]
+gap> MyJacobi(0,0);
+Error, <m> must be positive