A Short Equation in Reciprocals

Solution

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The equation is equivalent to z = xy/(x + y). Let d = gcd(x, y). Then

x = dm, y = dn, with gcd(m, n) = 1.

It follows that gcd(mn, m + n) = 1 so that

z = dmn / (m + n),

which implies (m + n) | d, i.e., d = k(m + n), k a positive integer.

Thus we obtain the solutions:

x = km(m + n), y = kn(m + n), z = kmn,

where the three parameters k, m, n are positive integers.

Remark

Assume a, b, c are positive integers with no common factor that satisfy 1/a + 1/b = 1/c, then a + b is a square!

Indeed, from the foregoing solution, k = 1, a = m(n + m), b = n(m + n), a + b = (m + n)².

Let positive integer a, b, c satisfy 1/a + 1/b = 1/c. Then a² + b² + c² is a square!

Indeed,

References

1. T. Andreescu, D. Andrica, I. Cucurezeanu, An Introduction to Diophantine Equations, Birkhäuser, 2010, pp. 22-23

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