132 lines
3.0 KiB
Go
132 lines
3.0 KiB
Go
/*
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Copyright © 2025 filifa
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This program is free software: you can redistribute it and/or modify
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it under the terms of the GNU General Public License as published by
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the Free Software Foundation, either version 3 of the License, or
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(at your option) any later version.
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This program is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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GNU General Public License for more details.
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You should have received a copy of the GNU General Public License
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along with this program. If not, see <http://www.gnu.org/licenses/>.
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*/
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package sieve
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// NOTE: these formulas come from https://web.archive.org/web/20151224013638/http://www.maa.org/sites/default/files/Walter_D22068._Stangl.pdf
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func qrPowersOfTwoCoprime(sieve []uint, n uint) {
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for q := uint(8); 2*q < n; q *= 2 {
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// q(2^n) = 2^(n-3) for n >= 3
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sieve[2*q] = 2 * sieve[q]
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}
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}
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func qrPowersOfOddPrimesCoprime(sieve []uint, p uint, n uint) {
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for q := p; p*q < n; q *= p {
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// q(p^n) = (p^n - p^(n-1)) / 2
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sieve[p*q] = (p*q - q) / 2
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}
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}
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func qrPowersOfTwo(sieve []uint, n uint) {
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k := 1
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for q := uint(1); 2*q < n; q *= 2 {
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if k%2 == 0 {
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// s(2^n) = (2^(n-1) + 4) / 3 for even n
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sieve[2*q] = (q + 4) / 3
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} else {
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// s(2^n) = (2^(n-1) + 5) / 3 for odd n
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sieve[2*q] = (q + 5) / 3
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}
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k += 1
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}
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}
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func qrPowersOfOddPrimes(sieve []uint, p uint, n uint) {
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k := 3
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for q := p * p; p*q < n; q *= p {
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if k%2 == 0 {
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// s(p^n) = (p^(n+1) + p + 2) / (2*(p+1)) for even n
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sieve[p*q] = (p*p*q + p + 2) / (2 * (p + 1))
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} else {
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// s(p^n) = (p^(n+1) + 2*p + 1) / (2*(p+1)) for odd n
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sieve[p*q] = (p*p*q + 2*p + 1) / (2 * (p + 1))
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}
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k += 1
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}
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}
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/*
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QuadraticResidues computes the number of quadratic residues modulo k for k=1 to n.
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see https://oeis.org/A000224 and https://oeis.org/A046073
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*/
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func QuadraticResidues(n uint, coprime bool, buflen uint) chan uint {
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sieve := make([]uint, n)
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for i := uint(0); i < n; i++ {
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sieve[i] = 1
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}
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ch := make(chan uint, buflen)
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go func() {
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defer close(ch)
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if coprime {
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sieveQRCoprime(sieve, n, ch)
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} else {
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sieveQR(sieve, n, ch)
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}
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}()
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return ch
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}
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func sieveQRCoprime(sieve []uint, n uint, ch chan uint) {
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for i := uint(0); i < n; i++ {
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if i == 0 || i == 1 || i == 4 || i == 6 || i == 8 || i == 12 || i == 24 || sieve[i] != 1 {
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ch <- sieve[i]
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continue
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}
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if i == 2 {
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qrPowersOfTwoCoprime(sieve, n)
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} else {
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// q(p) = (p - 1) / 2
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sieve[i] = (i - 1) / 2
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qrPowersOfOddPrimesCoprime(sieve, i, n)
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}
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updateMultiples(sieve, i, n, false)
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ch <- sieve[i]
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}
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}
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func sieveQR(sieve []uint, n uint, ch chan uint) {
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for i := uint(0); i < n; i++ {
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if i == 0 || i == 1 || sieve[i] != 1 {
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ch <- sieve[i]
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continue
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}
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if i == 2 {
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qrPowersOfTwo(sieve, n)
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} else {
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// s(p) = (p + 1) / 2
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sieve[i] = (i + 1) / 2
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if i*i < n {
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// s(p^2) = (p^2 - p + 2) / 2
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sieve[i*i] = (i*i - i + 2) / 2
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}
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qrPowersOfOddPrimes(sieve, i, n)
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}
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updateMultiples(sieve, i, n, false)
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ch <- sieve[i]
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}
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}
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