largest integer $x$ satisfying $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{w} = \frac{1}{13}$ when $x<y<z<w$ [closed]

What is the largest integer $x$ satisfying $$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{w}=\frac{1}{13}$$ considering that $x<y<z<w$.

How and which way to follow to solve above problem? Any insights?

I have a Python program which should do what I said in a comment above:

• Look for integer $x$ so that $1/x<1/13$ but $4/x>1/13$
• Look for integer $y>x$ so that $1/x+1/y<1/13$ but $1/x+3/y>1/13$
• Look for integer $z>y$ so that $1/x+1/y+1/z<1/13$ but $1/x+1/y+2/z>1/13$
• Calculate $w$ so that $1/x+1/y+1/z+1/w=1/13$ and that $w>z$
• Check if $w$ is an integer. If yes, output $x,y,z,w$.

The code is as follows:

from fractions import Fraction
import math

nsol = 0
sum4 = Fraction(1,13)
for x in range(math.floor(1/sum4)+1, math.ceil(4/sum4)):
    sum3 = sum4 - Fraction(1,x)
    for y in range(max(x+1, math.floor(1/sum3)+1), math.ceil(3/sum3)):
        sum2 = sum3 - Fraction(1,y)
        for z in range(max(y+1, math.floor(1/sum2)+1), math.ceil(2/sum2)):
            sum1 = sum2 - Fraction(1,z)
            w = math.floor(1/sum1)
            if w > z and Fraction(1,w) == sum1:
                print(x, y, z, w)
                nsol = nsol + 1

print(nsol, "solutions")

It prints out $4987$ solutions, out of which there are three with the biggest $x=39$:

39 40 60 104
39 42 56 104
39 52 60 65