move to lib

internal/lib/discreteLog.go

@@ -0,0 +1,79 @@
+/*
+Copyright © 2025 filifa
+
+This program is free software: you can redistribute it and/or modify
+it under the terms of the GNU General Public License as published by
+the Free Software Foundation, either version 3 of the License, or
+(at your option) any later version.
+
+This program is distributed in the hope that it will be useful,
+but WITHOUT ANY WARRANTY; without even the implied warranty of
+MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+GNU General Public License for more details.
+
+You should have received a copy of the GNU General Public License
+along with this program. If not, see <http://www.gnu.org/licenses/>.
+*/
+package lib
+
+import (
+	"errors"
+	"fmt"
+	"math/big"
+)
+
+// whyyyy doesn't math/big have a ceil functionnnn
+func ceilSqrt(x *big.Int) *big.Int {
+	z := new(big.Int).Sqrt(x)
+	s := new(big.Int).Exp(z, big.NewInt(2), nil)
+	if s.Cmp(x) != 0 {
+		z.Add(z, big.NewInt(1))
+	}
+
+	return z
+}
+
+// TODO: this can be extended to work with n, b not coprime
+// https://cp-algorithms.com/algebra/discrete-log.html
+func BabyStepGiantStep(n, b, x, order *big.Int) (*big.Int, error) {
+	z := new(big.Int).GCD(nil, nil, b, n)
+	if z.Cmp(big.NewInt(1)) != 0 {
+		return nil, fmt.Errorf("base %v and modulus %v are not coprime", b, n)
+	}
+
+	var m *big.Int
+	if order == nil {
+		// m = ceil(sqrt(n - 1))
+		z := big.NewInt(1)
+		z.Sub(n, z)
+		m = ceilSqrt(z)
+	} else {
+		m = ceilSqrt(order)
+	}
+
+	table := make(map[string]*big.Int)
+	for j := big.NewInt(1); j.Cmp(m) <= 0; j.Add(j, big.NewInt(1)) {
+		a := new(big.Int).Exp(b, j, n)
+		table[a.String()] = new(big.Int).Set(j)
+	}
+
+	// p = b^-m modulo n
+	p := new(big.Int).Neg(m)
+	p.Exp(b, p, n)
+
+	gamma := new(big.Int).Set(x)
+
+	for i := big.NewInt(0); i.Cmp(m) == -1; i.Add(i, big.NewInt(1)) {
+		j, ok := table[gamma.String()]
+		if ok {
+			i.Mul(i, m)
+			i.Add(i, j)
+			return i, nil
+		}
+
+		gamma.Mul(gamma, p)
+		gamma.Mod(gamma, n)
+	}
+
+	return nil, errors.New("no solution")
+}

internal/lib/divisor.go

@@ -0,0 +1,40 @@
+/*
+Copyright © 2025 filifa
+
+This program is free software: you can redistribute it and/or modify
+it under the terms of the GNU General Public License as published by
+the Free Software Foundation, either version 3 of the License, or
+(at your option) any later version.
+
+This program is distributed in the hope that it will be useful,
+but WITHOUT ANY WARRANTY; without even the implied warranty of
+MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+GNU General Public License for more details.
+
+You should have received a copy of the GNU General Public License
+along with this program. If not, see <http://www.gnu.org/licenses/>.
+*/
+package lib
+
+import (
+	"math/big"
+)
+
+// TODO: expand to work for different sigmas
+func DivisorSummatory(n *big.Int) *big.Int {
+	// employing Dirichlet's hyperbola method
+	sqrt := new(big.Int).Sqrt(n)
+
+	total := big.NewInt(0)
+	for x := big.NewInt(1); x.Cmp(sqrt) <= 0; x.Add(x, big.NewInt(1)) {
+		z := new(big.Int).Div(n, x)
+		total.Add(total, z)
+	}
+
+	total.Mul(total, big.NewInt(2))
+
+	sqrt.Exp(sqrt, big.NewInt(2), nil)
+	total.Sub(total, sqrt)
+
+	return total
+}

internal/lib/divisors.go

@@ -0,0 +1,75 @@
+/*
+Copyright © 2025 filifa
+
+This program is free software: you can redistribute it and/or modify
+it under the terms of the GNU General Public License as published by
+the Free Software Foundation, either version 3 of the License, or
+(at your option) any later version.
+
+This program is distributed in the hope that it will be useful,
+but WITHOUT ANY WARRANTY; without even the implied warranty of
+MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+GNU General Public License for more details.
+
+You should have received a copy of the GNU General Public License
+along with this program. If not, see <http://www.gnu.org/licenses/>.
+*/
+package lib
+
+func pow(base uint, exp uint) uint {
+	result := uint(1)
+	for exp > 0 {
+		if exp%2 == 1 {
+			result *= base
+		}
+
+		exp >>= 1
+		base *= base
+	}
+
+	return result
+}
+
+func updateMultiples(sieve []uint, x uint, p uint, n uint) {
+	for q := p; ; q *= p {
+		// sigma_x(a*b) = sigma_x(a) * sigma_x(b) if gcd(a,b) = 1
+		for i := 2 * q; i < n; i += q {
+			if i%(p*q) != 0 {
+				sieve[i] *= sieve[q]
+			}
+		}
+
+		if p*q >= n {
+			break
+		}
+
+		// sigma_x(p^k) = p^(kx) + sigma_x(p^(k-1))
+		sieve[p*q] = pow(p*q, x) + sieve[q]
+	}
+}
+
+func DivisorsSieve(n uint, x uint) chan uint {
+	sieve := make([]uint, n)
+	sieve[0] = 0
+	for i := uint(1); i < n; i++ {
+		sieve[i] = 1
+	}
+
+	ch := make(chan uint)
+	go func() {
+		for i := uint(0); i < n; i++ {
+			if i == 0 || i == 1 || sieve[i] != 1 {
+				ch <- sieve[i]
+				continue
+			}
+
+			sieve[i] = pow(i, x) + 1
+			updateMultiples(sieve, x, i, n)
+			ch <- sieve[i]
+		}
+
+		close(ch)
+	}()
+
+	return ch
+}

internal/lib/mobius.go

@@ -0,0 +1,48 @@
+/*
+Copyright © 2025 filifa
+
+This program is free software: you can redistribute it and/or modify
+it under the terms of the GNU General Public License as published by
+the Free Software Foundation, either version 3 of the License, or
+(at your option) any later version.
+
+This program is distributed in the hope that it will be useful,
+but WITHOUT ANY WARRANTY; without even the implied warranty of
+MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+GNU General Public License for more details.
+
+You should have received a copy of the GNU General Public License
+along with this program. If not, see <http://www.gnu.org/licenses/>.
+*/
+package lib
+
+func MobiusSieve(n uint) chan int {
+	sieve := make([]int, n)
+	for i := 0; i < int(n); i++ {
+		sieve[i] = i
+	}
+
+	ch := make(chan int)
+	go func() {
+		for i := 0; i < int(n); i++ {
+			if i == 0 || i == 1 || sieve[i] != i {
+				ch <- sieve[i]
+				continue
+			}
+
+			ch <- -1
+
+			for j := 2 * i; j < int(n); j += i {
+				if j%(i*i) == 0 {
+					sieve[j] = 0
+				}
+
+				sieve[j] /= -i
+			}
+		}
+
+		close(ch)
+	}()
+
+	return ch
+}

internal/lib/partitions.go

@@ -0,0 +1,52 @@
+/*
+Copyright © 2025 filifa
+
+This program is free software: you can redistribute it and/or modify
+it under the terms of the GNU General Public License as published by
+the Free Software Foundation, either version 3 of the License, or
+(at your option) any later version.
+
+This program is distributed in the hope that it will be useful,
+but WITHOUT ANY WARRANTY; without even the implied warranty of
+MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+GNU General Public License for more details.
+
+You should have received a copy of the GNU General Public License
+along with this program. If not, see <http://www.gnu.org/licenses/>.
+*/
+package lib
+
+import (
+	"math/big"
+)
+
+// given a slice where vals[i] = p_{k-1}(i) for a given k, nextPartition updates the slice so vals[i] = p_k(i) using the property that p_k(n) = p_k(n-k) + p_{k-1}(n)
+func nextPartition(k int, vals []*big.Int) {
+	for i := 1; i < len(vals); i++ {
+		if i-k >= 0 {
+			vals[i].Add(vals[i], vals[i-k])
+		}
+	}
+}
+
+func Partitions(n, k int) *big.Int {
+	if n < 0 {
+		return big.NewInt(0)
+	}
+
+	if k > n {
+		k = n
+	}
+
+	vals := make([]*big.Int, n+1)
+	for i := range vals {
+		vals[i] = big.NewInt(0)
+	}
+	vals[0] = big.NewInt(1)
+
+	for i := 1; i <= k; i++ {
+		nextPartition(i, vals)
+	}
+
+	return vals[n]
+}

internal/lib/primeOmega.go

@@ -0,0 +1,61 @@
+/*
+Copyright © 2025 filifa
+
+This program is free software: you can redistribute it and/or modify
+it under the terms of the GNU General Public License as published by
+the Free Software Foundation, either version 3 of the License, or
+(at your option) any later version.
+
+This program is distributed in the hope that it will be useful,
+but WITHOUT ANY WARRANTY; without even the implied warranty of
+MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+GNU General Public License for more details.
+
+You should have received a copy of the GNU General Public License
+along with this program. If not, see <http://www.gnu.org/licenses/>.
+*/
+package lib
+
+func primeOmegaUpdateMultiples(sieve []uint, p uint, n uint, multiplicity bool) {
+	for q := p; ; q *= p {
+		// omega(a*b) = omega(a) + omega(b) if gcd(a,b) = 1
+		for i := 2 * q; i < n; i += q {
+			if i%(p*q) != 0 {
+				sieve[i] += sieve[q]
+			}
+		}
+
+		if p*q >= n {
+			break
+		}
+
+		if multiplicity {
+			sieve[p*q] = 1 + sieve[q]
+		}
+	}
+}
+
+func PrimeOmegaSieve(n uint, multiplicity bool) chan uint {
+	sieve := make([]uint, n)
+	for i := uint(0); i < n; i++ {
+		sieve[i] = 0
+	}
+
+	ch := make(chan uint)
+	go func() {
+		for i := uint(0); i < n; i++ {
+			if i == 0 || i == 1 || sieve[i] != 0 {
+				ch <- sieve[i]
+				continue
+			}
+
+			sieve[i] = 1
+			primeOmegaUpdateMultiples(sieve, i, n, multiplicity)
+			ch <- sieve[i]
+		}
+
+		close(ch)
+	}()
+
+	return ch
+}

internal/lib/shoelace.go

@@ -0,0 +1,33 @@
+/*
+Copyright © 2025 filifa
+
+This program is free software: you can redistribute it and/or modify
+it under the terms of the GNU General Public License as published by
+the Free Software Foundation, either version 3 of the License, or
+(at your option) any later version.
+
+This program is distributed in the hope that it will be useful,
+but WITHOUT ANY WARRANTY; without even the implied warranty of
+MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+GNU General Public License for more details.
+
+You should have received a copy of the GNU General Public License
+along with this program. If not, see <http://www.gnu.org/licenses/>.
+*/
+package lib
+
+type Point struct {
+	X float64
+	Y float64
+}
+
+func Area(points []Point) float64 {
+	total := float64(0)
+	n := len(points)
+	for i, p := range points {
+		q := points[(i+1)%n]
+		total += (p.Y + q.Y) * (p.X - q.X)
+	}
+
+	return total / 2
+}

internal/lib/totient.go

@@ -0,0 +1,44 @@
+/*
+Copyright © 2025 filifa
+
+This program is free software: you can redistribute it and/or modify
+it under the terms of the GNU General Public License as published by
+the Free Software Foundation, either version 3 of the License, or
+(at your option) any later version.
+
+This program is distributed in the hope that it will be useful,
+but WITHOUT ANY WARRANTY; without even the implied warranty of
+MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+GNU General Public License for more details.
+
+You should have received a copy of the GNU General Public License
+along with this program. If not, see <http://www.gnu.org/licenses/>.
+*/
+package lib
+
+func TotientSieve(n uint) chan uint {
+	totients := make([]uint, n)
+	totients[0] = 0
+	totients[1] = 1
+	for i := uint(2); i < n; i++ {
+		totients[i] = i - 1
+	}
+
+	ch := make(chan uint)
+	go func() {
+		for i := uint(0); i < n; i++ {
+			ch <- totients[i]
+			if i == 0 || i == 1 || totients[i] != i-1 {
+				continue
+			}
+
+			for j := uint(2 * i); j < n; j += i {
+				totients[j] -= totients[j] / i
+			}
+		}
+
+		close(ch)
+	}()
+
+	return ch
+}