Find all possible triplets (x, y, z) of positive integers that satisfy this equation:

(1+1/x)*(1+1/y)*(1+1/z) = 5

Prove that these are the only possible triplets in conformity with the given conditions.

Note: Computer program solutions are welcome, but an analytic solution is preferred.

(1, 1, 4) in any order seems to be the only triplet that works.

Suppose x=y=z and they do not need to be integers:
then (x+1)^3 = 5 x^3
so x = 1 / (5^(1/3) - 1) =~ 1.408 which will be the geometric mean of (x,y,z)

For large values of x,y,z the RHS tends toward 3 which is too small.
Even if x=y=z=2, the RHS is 3.375 and is too small.
What if (x,y,z) = (1,2,2) --> RHS = 4.5 still too small
Therefore two of (x,y,z) must be 1.
wlog, we have (1,1,z) --> (2)*(2)*(z+1)/z = 5
(z+1)/z = 5/4
z = 4

------------------
big = 1000
for x in range(1,big):
    for y in range(x, big):
        for z in range(y, big):
            if (1+1/x)*(1+1/y)*(1+1/z) == 5:
                print(x,y,z)

Posted by Larry on 2023-06-11 20:06:40