difference_of_two_rectangles_solutions.sf

#!/usr/bin/ruby

# Author: Trizen
# Date: 18 March 2022
# https://github.com/trizen

# Represent an integer as a difference of two rectangles:
#
#   n = a*b - c*d
#
# with a,b,c,d > 1 and a != b and c != d.

# This algorithm works only for integers that can also be represented as a difference of two squares.

func difference_of_two_squares_solutions(n) {

    n.divisors.map {|d|

        break if (d*d >= n)

        var a = d
        var b = n/d

        (a+b).is_even || next

        var x = (a + b)/2
        var y = (b - a)/2

        [x, y]
    }
}

func difference_of_two_rectangles_solutions(n) {

    var solutions = []

    difference_of_two_squares_solutions(n).each_2d {|x,y|

        var ab = []
        var cd = []

        difference_of_two_squares_solutions(x**2).each_2d {|a,b|
            ab << [a+b, a-b] if (a-b > 1)
        }

        difference_of_two_squares_solutions(y**2).each_2d {|c,d|
            cd << [c+d, c-d] if (c-d > 1)
        }

        [ab,cd].cartesian {|x,y|
            solutions << [x..., y...]
        }
    }

    return solutions.sort
}

var n = 420

difference_of_two_rectangles_solutions(n).each_2d {|a,b,c,d|
    say "#{a*b - c*d} = #{a}*#{b} - #{c}*#{d}"
}

__END__
420 = 242*2 - 16*4
420 = 242*2 - 32*2
420 = 338*2 - 32*8
420 = 338*2 - 64*4
420 = 338*2 - 128*2
420 = 722*2 - 64*16
420 = 722*2 - 128*8
420 = 722*2 - 256*4
420 = 722*2 - 512*2
420 = 5618*2 - 208*52
420 = 5618*2 - 338*32
420 = 5618*2 - 416*26
420 = 5618*2 - 676*16
420 = 5618*2 - 1352*8
420 = 5618*2 - 2704*4
420 = 5618*2 - 5408*2