The Research On Indeterminate Equation Involving Some Arithmetic Function

The research of indeterminate equation is the topic of people’s interest, especially the indeterminate equation involving arithmetic functions. Many experts and scholars have made deep exploration and research on these issues and obtained a lot of achievements with important academic value. Using elementary number theory methods to study the equation involving arithmetic functions and particular integer sequences, some positive integer solutions are obtained.Firstly, the main purpose is to discuss the solutions of equation involving Smarandache primitive function and particular integer sequences. Combined Smarandache primitive function with triangular numbers and pentagonal numbers, two equations are obtained. Using elementary number theory methods, the positive integer solutions of these equations are obtained.Secondly, the main purpose is to discuss the solutions of equation involving Pell sequence and arithmetic functions. Combined Euler function, the sum of divisors function and Smarandache function with Pell sequence and Pell- Lucas sequence, a series of equations are obtained. Using elementary number theory methods and the property of Pell sequence and Pell- Lucas sequence, the relevant results are obtained.Thirdly, the main purpose is to discuss the necessary and sufficient condition of the indeterminate equation x~3-5~3=3py~2 has positive integer solution with gcd(x,y)=1 by using elementary number theory methods.Finally, the solutions of equations involving arithmetic function and particular sequences are concluded and some issues which need further research are pointed out.