All solutions of the Diophantine equation 1/x+1/y+1/z=u/(u+2)

In the history of mathematics, many mathematical researchers have investigated the Diophantine equation in the form , where and are positive integers. Without loss of generality, we may assume that . This Diophantine equation, also known as the Egyptian fraction equation of length 3, is to write the fraction as a sum of three fractions with the numerator being one and the denominators being different positive integers. Examples of research such as, in 2021, Sandor and Atanassov studied and found that the Diophantine equation has forty-four positive integer solutions. In this paper, we will study and find the complete positive integer solutions of the Diophantine equation , by using elementary methods of number theory and computer calculations. In the process, we can see that . Then, we will consider separately the value of a positive integer in nine cases. The first case is impossible. For the second and third cases, we will separate to consider the value of . For the remaining cases, we will separate to consider the value of . The research results showed that all positive integer solutions of the Diophantine equation are eighty-seven positive integer solutions. Moreover, from the steps to find the above positive integer solutions, we expect that it can be used to find the complete positive integer solutions of the Diophantine equation , where is a positive integer with .