Investigation of one class of diophantine equations - Bondarenko, A. V. (Ukrains’kyi Matematychnyi Zhurnal, 2000)
We consider the problem of existence of solutions of the equation \(XY + YZ + ZX = m\) in natural numbers for differentm∈N. We prove that this equation possesses solutions in natural numbers form=a 2+5,a∈Z, and does not have solutions ifm=4p 2,p∈N, andp is not divisible by 3. We also prove that, forn≥12, the equation b_1 b_2 + b_2 b_3 + … + b_n - 1 b_n + b_n b_1 = m possesses solutions in natural numbers if and only ifm≥n,m∈N.