What Are Other Solutions to the Equation 1/x + 1/y = 1/z for Integer Values?

[murshid_islam]

what are the solutions of the equation \frac{1}{x} + \frac{1}{y} = \frac{1}{z} where x, y, z \in \mathbb{Z}

i can see that x = y = 2z is a set of solutions. is there any other? and how do i find those?

thanks in advance.

 

[D H]

What is the multiplicative inverse of both sides of the equation?

 

[murshid_islam]

What is the multiplicative inverse of both sides of the equation?

i am sorry. i didn't understand how that would help. i got z = \frac{xy}{x+y}

 

[Werg22]

Transform the expression as such

\frac{x+y}{xy} = \frac{1}{z}

\frac{xy}{x+y} = zx and y can obviously not be both odd. Also, if there's no common factor between x and y, then z is never an integer. Let's say a common factor between x and y is a. We have \frac{xy}{a(\frac{x}{a}+\frac{y}{a})} = z

In turn x/a and y/a need to have common factors for z to be an integer. If we have y > x and keep on and on, we eventually get to the point where

a_{1}*a_{2}* a_{3} *a_{4} *a_{5}... = x

We conclude that y and x must have the same prime factors and that the number of times that a prime factor reoccurs in x is matched or surpassed by y. Now say y = kx, we have

\frac{xy}{x(1+\frac{y}{x})}

\frac{y}{1+\frac{y}{x}}

\frac{kx}{1+k}

We can immediately see that 1 + k has to be a factor of x if z is to be an integer. Knowing that, we can construct an infinite number of pairs that satisfy the original expression; we need an y such as y = kx, and a x such that k+1 is one of its factor. For example, say k = 2, we can have x = (2+1)*7*5 and y = 2*(2+1)*7*5. So a solution is such as, if C and k are any positive integers, x = C(k+1), y = Ck(k+1)

 

[murshid_islam]

sorry, but i didn't understand some of your reasonings. for example,

Also, if there's no common factor between x and y, then z is never an integer.

why is that?

 

[Werg22]

Just a hunch I could be wrong. But the solution is valid for the case x and y have common factors. Also the kx/1+k thing is a hunch too.

 

[robert Ihnot]

There is something called, "Egyptian fractions." These are all the inverse of integers. It seemed that the Ancient Egyptians did not understand fractions except in the form of 1/N, and duplicates were not used. This system was even used by the Greeks for awhile, and results in very complicated things. Here is a simple formula that is useful: 1/N = 1/(N+1) + 1/((N+1)(N)).

Like say, we have the matter of 1= 1/3+1/3+1/3. We can expand this as: 1=1/3+1/4+1/12 +1/4+1/12 and finally: 1/3+1/4+1/12+1/5+1/20+1/13+1/156.

This then would have satisfied the Egyptian mathematicians of the day.

Some people believe there was SOME SENSE in this system. http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fractions/egyptian.html#rhind

For example, suppose 5 sacks of grain are to be divided up by 8 people. Well we give each person 1/2 sack and then we take the remaining sack and divide it into 8 parts. Hooray! 1/2+1/8 = 5/8.

 

[murshid_islam]

Werg22's method in the post#4 seemed really complicated and i didn't understand half of it. can anyone please check if there is any problem with this method:

\frac{1}{x} + \frac{1}{y} = \frac{1}{z}

if i put x = z+a and y = z+b, (where and b are integers) the equation becomes:

\frac{1}{z+a} + \frac{1}{z+b} = \frac{1}{z}

\frac{2z + a + b}{z^2 + az + bz + ab} = \frac{1}{z}

2z^2 + az + bz = z^2 + az + bz + ab

z^2 = ab

now we just have to factor z². for example if the equation was
1/x + 1/y = 1/2007
we would put x = 2007+a and y = 2007+b and eventually get (2007)² = ab. and then we just have to factor (2007)² to find a and b.

 

[Integral]

IIRC you can do a coordinate rotation of \frac {\pi} 4 to see that this is a hyperbola. (This is just the thin lens equation with different variable names)

 

[uart]

Werg22's method in the post#4 seemed really complicated and i didn't understand half of it. can anyone please check if there is any problem with this method:

\frac{1}{x} + \frac{1}{y} = \frac{1}{z}

if i put x = z+a and y = z+b, (where and b are integers) the equation becomes:

\frac{1}{z+a} + \frac{1}{z+b} = \frac{1}{z}

\frac{2z + a + b}{z^2 + az + bz + ab} = \frac{1}{z}

2z^2 + az + bz = z^2 + az + bz + ab

z^2 = ab

now we just have to factor z². for example if the equation was
1/x + 1/y = 1/2007
we would put x = 2007+a and y = 2007+b and eventually get (2007)² = ab. and then we just have to factor (2007)² to find a and b.

That's a good solution Murshid.

I had another solution that I didn't post. I won't post the details but I followed a similar path to werg to get it. In particular I assummed that x and y must have a common factor (something that at the time I couldn't really justify).

My solution was basically this. Let jk be factor of z. That is, z = ajk. Then
x = aj (j+k)
y = ak (j+k)
are solutions to your equation (1/x + 1/y = 1/z).

I like your solution a lot better. It was derived without the assumption of x and y sharing a common factor, though interestingly it can be used to prove that x and y must indeed have this common factor.

A rough proof that x and y (as per your solution) contain a non trivial common factor is as follows.

Let x = a + z and y = b + z,
where a and b are any factor pair of z^2

Since perfects squares must have all their prime factors in pairs, and since ab=z^2, then either a and b are themselves both perfect squares or they contain at least one common factor.

Case 1 : a and b are not perfect squares.
As above, a and b contain at least one common prime factor. Also z contains all the prime factors of z^2, so z shares the same common factors as a and b above (since z^2=ab). These common factors are thereofore also common factors of x and y.

Case 2 : a and b are both perfect squares.
In this case we just rewrite x and y as,
x = sqrt(a) (sqrt(a) + sqrt(b))
y = sqrt(b) (sqrt(a) + sqrt(b))
So clearly x and y have a common factor.

So the original hunch that x and y must have a common factor was indeeed correct.