python – 为什么我天真的阿特金斯筛选实施排除5？

从技术上讲,我知道为什么要排除5;在n = 4 * x ^ 2 y ^ 2的步骤中,当x == 1且y == 1时n == 5.这仅发生一次,因此5从素数转换为非素数并且从不翻转.

为什么我的算法与维基百科上的算法不匹配？虽然我做了一些表面调整(例如,在每次迭代中只计算一次x ^ 2,在第一个等式中使用时存储mod(n,12)的值等),但它们不应该改变逻辑.算法.

我在several discussions related阅读了Atkin的Sieve,但我不知道在我的实现中产生问题的区别是什么.

Python代码：

def atkin1(limit):
    res = [0] * (limit + 1)
    res[2] = 1
    res[3] = 1
    res[5] = 1

    limitSqrt = int(math.sqrt(limit))
    for x in range(1,limitSqrt+1):
        for y in range(1,limitSqrt+1):
            x2 = x**2
            y2 = y**2
            n = 4*x2 + y2
            if n == 5:
                print('debug1')
            nMod12 = n % 12
            if n <= limit and (nMod12 == 1 or nMod12 == 5):
                res[n] ^= 1

            n = 3*x2 + y2
            if n == 5:
                print('debug2')
            if n <= limit and (n % 12 == 7):
                res[n] ^= 1

            if x > y:
                n = 3*x2 - y2
                if n == 5:
                    print('debug3')
                if n <= limit and (n % 12 == 11):
                    res[n] ^= 1

    ndx = 5
    while ndx <= limitSqrt:
        m = 1
        if res[ndx]:
            ndx2 = ndx**2
            ndx2Mult =m * ndx2
            while ndx2Mult < limit:
                res[ndx2Mult] = 0
                m += 1
                ndx2Mult = m * ndx2
        ndx += 1

    return res

MATLAB代码

function p = atkin1(limit)

% 1. Create a results list,filled with 2,3,and 5
res = [0,1,1];

% 2. Create a sieve list with an entry for each positive integer; all entries of
% this list should initially be marked nonprime (composite).
res = [res,zeros(1,limit-5)];

% 3. For each entry number n in the sieve list,with modulo-sixty remainder r:

limitSqrt = floor(sqrt(limit));
for x=1:limitSqrt
    for y=1:limitSqrt
        x2 = x^2;       y2 = y^2;

        % If r is 1,13,17,29,37,41,49,or 53,flip the entry for each
        % possible solution to 4x^2 + y^2 = n.
        n = 4*x2 + y2;
        nMod12 = mod(n,12); 
        if n <= limit && (nMod12 == 1 || nMod12 == 5)
            res(1,n) = ~res(1,n);
        end

        % If r is 7,19,31,or 43,flip the entry for each possible solution
        % to 3x^2 + y^2 = n.
        n = 3*x2 + y2;
        if n <= limit && mod(n,12) == 7
            res(1,n);
        end

        % If r is 11,23,47,or 59,flip the entry for each possible solution
        % to 3x^2 - y^2 = n when x > y.
        if x > y
            n = 3*x2 - y2;
            if n <= limit && mod(n,12) == 11
                res(1,n);
            end
        end

        % If r is something else,ignore it completely.
    end
end

   % 4. Start with the lowest number in the sieve list.
ndx = 5;
while ndx < limitSqrt
    m = 1;
    if res(ndx)
        % 5. Take the next number in the sieve list still marked prime.
        % 6. Include the number in the results list.
        % 7. Square the number and mark all multiples of that square as nonprime.
        ndx2 = ndx^2;
        ndx2Mult = m * ndx2;
        while ndx2Mult < limit
            res(ndx2Mult) = 0;
            m = m + 1;
            ndx2Mult = m * ndx2;
        end
    end

    % 8. Repeat steps five through eight.
    ndx = ndx + 1;
end

p = find(res); % Find the indexes of nonzerogo
end

Wikipedia pseudocode

// arbitrary search limit
limit ← 1000000         

// initialize the sieve
is_prime(i) ← false,? i ∈ [5,limit] 

// put in candidate primes: 
// integers which have an odd number of
// representations by certain quadratic forms
for (x,y) in [1,√limit] × [1,√limit]:
    n ← 4x2+y2
    if (n ≤ limit) and (n mod 12 = 1 or n mod 12 = 5):
        is_prime(n) ← ?is_prime(n)
    n ← 3x2+y2
    if (n ≤ limit) and (n mod 12 = 7):
        is_prime(n) ← ?is_prime(n)
    n ← 3x2-y2
    if (x > y) and (n ≤ limit) and (n mod 12 = 11):
        is_prime(n) ← ?is_prime(n)

// eliminate composites by sieving
for n in [5,√limit]:
    if is_prime(n):
        // n is prime,omit multiples of its square; this is
        // sufficient because composites which managed to get
        // on the list cannot be square-free
        is_prime(k) ← false,k ∈ {n2,2n2,3n2,...,limit} 

print 2,3
for n in [5,limit]:
    if is_prime(n): print n