An Improved Solution of the Diophantine Equation $\sum_{i=1}^k 1/X_i=1$ in Dictinct Integers of the form $X_ii\in P^\alphaq^\beta$
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چکیده
Abstract. Solutions of the title equation are discussed, and in particular when all xi are even, p < q are primes, and α,β are positive integers. The author [7] has shown that when all xi are distinct, even, and of the form xi p q, then the title equation has a solution with k = 85. Our main result consists of an improved solution to the title equation when xi are even, xi p q and k = 67. Moreover, the solution in [7] contains seven distinct odd primes, whereas in the current result this number is reduced to the first six odd primes. Some modifications of the equation and questions are also presented.
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