748 Upside down Diophantine equation.sf

#!/usr/bin/ruby

# Daniel "Trizen" Șuteu
# Date: 27 February 2021
# https://github.com/trizen

# Upside down Diophantine equation
# https://projecteuler.net/problem=748

# The problem asks for integer solutions to:
#   1/x^2 + 1/y^2 = 13/z^2

# It is easy to see that:
#   (x^2 + y^2)/13 = v^4, for some integer v.

# Multiplying both sides by 13, we have:
#   x^2 + y^2 = v^4 * 13

# By finding integer solutions (x,y) to the above equation, we can then compute `z` as:
#   z = sqrt((x^2 * y^2 * 13)/(x^2 + y^2))
#   z = sqrt((x^2 * y^2) / v^4)

# We need to iterate over 1 <= v <= sqrt(N/3).

func sum_of_two_squares_solutions(n) is cached {

    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.append([p, 1])
            }
        }
        else {                             # p = 1 (mod 4)
            prod1 *= p**e
            prime_powers.append([p, e])
        }
    }

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

    # All the solutions to the congruence: x^2 = -1 (mod prod1)
    var square_roots = gather {
        gather {
            for p,e in (prime_powers) {
                var pp = p**e
                var r = sqrtmod(-1, pp)
                take([[r, pp], [pp - r, pp]])
            }
        }.cartesian { |*a|
            take(Math.chinese(a...))
        }
    }

    var solutions = []

    for r in (square_roots) {

        var s = r
        var q = prod1

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

        solutions.append([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]   }
}

func S(N) {

    var total = 0

    for v in (1 .. floor(sqrt(N/3))) {

        var solutions = sum_of_two_squares_solutions(13 * v**4) || next

        for x,y in (solutions) {

            y <= N || next

            var z = (x*x * y*y)/(v**4)

            if (z.is_square) {

                z = z.isqrt
                z <= N || next

                if (gcd(x,y,z) == 1) {
                    say [x,y,z,v]
                    total += [x,y,z].sum
                    assert_eq(1/x**2 + 1/y**2, 13/z**2)
                }
            }
        }
    }

    return total
}

assert_eq(S(1e2), 124)
assert_eq(S(1e3), 1470)
assert_eq(S(1e5), 2340084)

var total = S(1e16)
say "Total: #{total} -> #{total % 1e9}"