feat(tricks): primitive roots

Signed-off-by: Pouyan Heyratpour <[email protected]>

tricks/mathx/arithmetic.go

@@ -58,6 +58,26 @@ func PrimeFactors(n int) []int {
 	return ff[:len(ff):len(ff)]
 }
 
+// NextPrime returns next prime number greater than the given number (n)
+func NextPrime(n int) int {
+	if n < 1 {
+		n = 1
+	}
+
+	return nextPrime(n)
+}
+
+// nextPrime returns next prime number greater than the given number (n)
+func nextPrime(n int) int {
+	for {
+		n++
+
+		if IsPrime(n) {
+			return n
+		}
+	}
+}
+
 // CoprimesInRange returns list of all natural numbers in closed range [from, to]
 // which are coprime to the given number (n)
 func CoprimesInRange(n, from, to int) []int {
@@ -76,22 +96,31 @@ func MinorCoprimes(n int) []int {
 	return CoprimesInRange(n, 1, n-1)
 }
 
-// NextPrime returns next prime number greater than the given number (n)
-func NextPrime(n int) int {
-	if n < 1 {
-		n = 1
+// PrimitiveRoots returns list of all natural numbers which are less than the given number (n) and
+// primitive root modulo (n)
+func PrimitiveRoots(n int) map[int][]int {
+	cpp := CoprimesInRange(n, 2, n-1)
+	if len(cpp) == 0 {
+		return map[int][]int{1: {1}}
 	}
 
-	return nextPrime(n)
-}
+	prr := make(map[int][]int)
+	for _, cp := range cpp {
+		mm := make([]int, 0, len(cpp)+1)
+		m := 1
+		for i := 1; i < n; i++ {
+			m = (m * cp) % n
+			mm = append(mm, m)
 
-// nextPrime returns next prime number greater than the given number (n)
-func nextPrime(n int) int {
-	for {
-		n++
+			if m == 1 {
+				break
+			}
+		}
 
-		if IsPrime(n) {
-			return n
+		if len(mm) == len(cpp)+1 {
+			prr[cp] = mm[:len(mm):len(mm)]
 		}
 	}
+
+	return prr
 }

tricks/mathx/arithmetic_test.go

@@ -120,7 +120,7 @@ func TestIsPrime(t *testing.T) {
 	assert.False(t, mathx.IsPrime(-7))
 }
 
-func TestPrimes(t *testing.T) {
+func TestPrimeFactors(t *testing.T) {
 	assert.Equal(t, []int{2, 3}, mathx.PrimeFactors(6))
 	assert.Equal(t, []int{2, 5}, mathx.PrimeFactors(10))
 	assert.Equal(t, []int{2, 3, 5}, mathx.PrimeFactors(30))
@@ -162,16 +162,6 @@ func TestPrimes(t *testing.T) {
 	assert.Equal(t, []int{}, mathx.PrimeFactors(-127))
 }
 
-func TestCoprimesInRange(t *testing.T) {
-	assert.Equal(t, []int{3, 7}, mathx.CoprimesInRange(10, 2, 7))
-	assert.Equal(t, []int{7, 9, 11, 13, 17, 19}, mathx.CoprimesInRange(10, 5, 20))
-}
-
-func TestMinorCoprimes(t *testing.T) {
-	assert.Equal(t, []int{1, 5}, mathx.MinorCoprimes(6))
-	assert.Equal(t, []int{1, 3, 7, 9}, mathx.MinorCoprimes(10))
-}
-
 func TestNextPrime(t *testing.T) {
 	assert.Equal(t, 2, mathx.NextPrime(0))
 	assert.Equal(t, 2, mathx.NextPrime(1))
@@ -188,3 +178,51 @@ func TestNextPrime(t *testing.T) {
 	assert.Equal(t, 2, mathx.NextPrime(-5))
 	assert.Equal(t, 2, mathx.NextPrime(-100))
 }
+
+func TestCoprimesInRange(t *testing.T) {
+	assert.Equal(t, []int{3, 7}, mathx.CoprimesInRange(10, 2, 7))
+	assert.Equal(t, []int{7, 9, 11, 13, 17, 19}, mathx.CoprimesInRange(10, 5, 20))
+}
+
+func TestMinorCoprimes(t *testing.T) {
+	assert.Equal(t, []int{1, 5}, mathx.MinorCoprimes(6))
+	assert.Equal(t, []int{1, 3, 7, 9}, mathx.MinorCoprimes(10))
+}
+
+func TestPrimitiveRoots(t *testing.T) {
+	assert.Equal(t, map[int][]int{
+		1: {1},
+	}, mathx.PrimitiveRoots(2))
+
+	assert.Equal(t, map[int][]int{
+		2: {2, 1},
+	}, mathx.PrimitiveRoots(3))
+
+	assert.Equal(t, map[int][]int{
+		3: {3, 2, 6, 4, 5, 1},
+		5: {5, 4, 6, 2, 3, 1},
+	}, mathx.PrimitiveRoots(7))
+
+	assert.Equal(t, map[int][]int{
+		3: {3, 9, 7, 1},
+		7: {7, 9, 3, 1},
+	}, mathx.PrimitiveRoots(10))
+
+	assert.Equal(t, map[int][]int{
+		3:  {3, 9, 10, 13, 5, 15, 11, 16, 14, 8, 7, 4, 12, 2, 6, 1},
+		5:  {5, 8, 6, 13, 14, 2, 10, 16, 12, 9, 11, 4, 3, 15, 7, 1},
+		6:  {6, 2, 12, 4, 7, 8, 14, 16, 11, 15, 5, 13, 10, 9, 3, 1},
+		7:  {7, 15, 3, 4, 11, 9, 12, 16, 10, 2, 14, 13, 6, 8, 5, 1},
+		10: {10, 15, 14, 4, 6, 9, 5, 16, 7, 2, 3, 13, 11, 8, 12, 1},
+		11: {11, 2, 5, 4, 10, 8, 3, 16, 6, 15, 12, 13, 7, 9, 14, 1},
+		12: {12, 8, 11, 13, 3, 2, 7, 16, 5, 9, 6, 4, 14, 15, 10, 1},
+		14: {14, 9, 7, 13, 12, 15, 6, 16, 3, 8, 10, 4, 5, 2, 11, 1},
+	}, mathx.PrimitiveRoots(17))
+
+	prr49 := mathx.PrimitiveRoots(49)
+	for _, pra := range prr49 {
+		for _, prb := range prr49 {
+			assert.ElementsMatch(t, pra, prb)
+		}
+	}
+}