Substituting: x = 2a, y = 2b, and z = 2c, in the original equation, one obtains:
2^2a + 2^5b = 2^3c

Considering 2^2a + 2^5b = 2^3c in conjunction with the identity
2^(k-1) + 2^(k-1) = 2^k
generates the equation :
2a + 1 = 5b + 1 = 3c

Since, the minimum possible positive integral solution to (#) is (a,b,c)=(10,4,7), by way of application of Chinese Remainder Theorem to (#), we obtain the following parametric relationship:
(a,b,c) = (15*s + 10, 6*s + 4, 10*s + 7)---------(#), so that
(#) generates infinite number of triplets (a,b,c) corresponding to integral s (positive 0r negative)

Since, by assumption x=2^a; y=2^b and z = 2^c; it immediately follows that there exists an infinite number of solutions to the equation under reference.
For example, for s= 1, 2, 3, 4, 5, 6 and 7 we obtain seven triplets:

NOTE: The parametric relationship in (#) may correspond to one of the numerous parametric solutions for the problem under reference. However, the relationship (#) alone is sufficient to prove that the total number of solutions to the problem is indeed infinite.

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