sum_of_two_squares_solutions_tonelli-shanks.sf

#!/usr/bin/ruby

# Daniel "Trizen" Șuteu
# Date: 26 October 2017
# https://github.com/trizen

# A recursive algorithm for finding all the non-negative integer solutions to the equation:
#   a^2 + b^2 = n
# for any given positive integer `n` for which such a solution exists.

# Example:
#   99025 = 41^2 + 312^2 = 48^2 + 311^2 = 95^2 + 300^2 = 104^2 + 297^2 = 183^2 + 256^2 = 220^2 + 225^2

# Blog post:
#   https://trizenx.blogspot.com/2017/10/representing-integers-as-sum-of-two.html

# This algorithm is efficient when the factorization of `n` can be computed.

# See also:
#   https://oeis.org/A001481

func tonelli(n, p) {

    var q = p-1
    var s = valuation(q, 2)

    s == 1 && return powmod(n, (p + 1) >> 2, p)

    q >>= s

    var c = powmod(2 ..^ p -> first {|z| legendre(z, p) == -1 }, q, p)
    var r = powmod(n, (q + 1) >> 1, p)
    var t = powmod(n, q, p)

    while (!p.divides(t - 1)) {

        var b = 1
        var t2 = (t*t % p)

        for i in (1 ..^ s) {
            if (p.divides(t2 - 1)) {
                b = powmod(c, 1 << (s - i - 1), p)
                s = i
                break
            }
            t2 = (t2*t2 % p)
        }

        r = (r*b % p)
        c = (b*b % p)
        t = (t*c % p)
    }

    return r
}

func sqrt_mod_n(a, n) is cached {

    a = (a % n)

    kronecker(a, n) == 1 || return []

    if ((n & (n - 1)) == 0) {       # n is a power of 2

        if (a % 8 == 1) {

            var k = n.valuation(2)

            k == 1 && return [1]
            k == 2 && return [1, 3]
            k == 3 && return [1, 3, 5, 7]

            if (a == 1) {
                return [1, (n>>1) - 1, (n>>1) + 1, n - 1]
            }

            return gather {
                for s in (sqrt_mod_n(a, n >> 1)) {
                    var i = (((s*s - a) >> (k - 1)) % 2)
                    var r = (s + (i << (k - 2)))
                    take(r, n - r)
                }
            }.uniq
        }

        return []
    }

    if (n.is_prime) {               # n is a prime
        return gather {
            var r = tonelli(a, n)
            take(r, n - r)
        }
    }

    var pe = n.factor_exp           # factorize `n` into prime powers

    if (pe.len == 1) {              # `n` is an odd prime power

        var p = pe[0][0]
        var k = pe[0][1]

        kronecker(a, p) == 1 || return []

        var (r1, r2) = with( tonelli(a, p) ) { |r|
            (r, n - r)
        }

        var pk = p
        var pi = p*p

        (k-1).times {
            var x = r1
            var y = (invmod(2, pk) * invmod(x, pk))

            r1 = ((pi + x - y*(x*x - a + pi)) % pi)
            r2 = (pi - r1)

            pk *= p
            pi *= p
        }

        return [r1, r2]
    }

    var solutions = []

    for p,e in (pe) {
        var m = p**e
        var r = sqrt_mod_n(a, m)
        solutions << r.map {|r0| [r0, m] }
    }

    gather {
        solutions.cartesian {|*a|
            take(Math.chinese(a...))
        }
    }.uniq
}

func sum_of_two_squares_solutions(n) is cached {

    n  < 0 && return []
    n == 0 && return [[0, 0]]

    var prod1 = 1
    var prod2 = 1

    var prime_powers = []

    for p,e in (n.factor_exp) {
        if (p % 4 == 3) {                  # p = 3 (mod 4)
            e.is_even || return []         # power must be even
            prod2 *= p**(e >> 1)
        }
        elsif (p == 2) {                   # p = 2
            if (e.is_even) {               # power is even
                prod2 *= p**(e >> 1)
            }
            else {                         # power is odd
                prod1 *= p
                prod2 *= p**((e - 1) >> 1)
                prime_powers << [p, 1]
            }
        }
        else {                             # p = 1 (mod 4)
            prod1 *= p**e
            prime_powers << [p, e]
        }
    }

    prod1 == 1 && return [[prod2, 0]]
    prod1 == 2 && return [[prod2, prod2]]

    var solutions = []

    for r in (sqrt_mod_n(-1, prod1)) {

        var s = r
        var q = prod1

        while (s*s > prod1) {
            (s, q) = (q % s, s)
        }

        solutions << [prod2 * s, prod2 * (q % s)]
    }

    for p,e in (prime_powers) {
        for (var i = e%2; i < e; i += 2) {

            var sq = p**((e - i) >> 1)
            var pp = p**(e - i)

            solutions += __FUNC__(prod1 / pp).map { |pair|
                pair.map {|r| sq * prod2 * r }
            }
        }
    }

    solutions.map     {|pair| pair.sort } \
             .uniq_by {|pair| pair[0]   } \
             .sort_by {|pair| pair[0]   }
}

50.times {
    var n = 1e10.irand
    var solutions = sum_of_two_squares_solutions(n) || next
    say %Q(#{n} = #{solutions.map {|a| "#{a[0]}^2 + #{a[1]}^2" }.join(' = ') })
}

assert_eq(sum_of_two_squares_solutions(2025), [[0, 45], [27, 36]])
assert_eq(sum_of_two_squares_solutions(164025), [[0, 405], [243, 324]])
assert_eq(sum_of_two_squares_solutions(99025), [[41, 312], [48, 311], [95, 300], [104, 297], [183, 256], [220, 225]])

assert_eq(
    -10 .. 160 -> grep { sum_of_two_squares_solutions(_).len > 0 },
    %n[0, 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25, 26, 29, 32, 34, 36, 37, 40, 41, 45, 49, 50, 52, 53, 58, 61, 64, 65, 68, 72, 73, 74, 80, 81, 82, 85, 89, 90, 97, 98, 100, 101, 104, 106, 109, 113, 116, 117, 121, 122, 125, 128, 130, 136, 137, 144, 145, 146, 148, 149, 153, 157, 160]
)

assert_eq(
    sum_of_two_squares_solutions(11392163240756069707031250),
    [[39309472125, 3374998963875], [216763660575, 3368260197225], [477329304375, 3341305130625], [729359177085, 3295481517405], [735019741071, 3294223614297], [907262616645, 3251005657515], [982736803125, 3228992353125], [1151205969375, 3172835964375], [1224793301193, 3145162095999], [1393801568775, 3074000720175], [1622919634875, 2959441687125], [1847545189875, 2824666354125], [1993551800625, 2723584854375], [2056446956025, 2676413487825], [2194367046795, 2564549961435], [2198769707673, 2560776252111], [2386646521875, 2386646521875]]
)