On negative Pell equations: Solvability and unsolvability in integers

Hsin-Te Chiang, Institute for Computational and Modeling Science, National Tsing Hua University, Mei-Ru Ciou, Chia-Ling Tsai, Yuh-Jenn Wu, Chiun-Chang Lee, Institute for Computational and Modeling Science, National Tsing Hua University, Institute for Computational and Modeling Science, National Tsing Hua University, Chung Yuan Christian University, Institute for Computational and Modeling Science, National Tsing Hua University
2018 Notes on Number Theory and Discrete Mathematics  
Solvability criteria of negative Pell equations x 2 − dy 2 = −1 have previously been established via calculating the length for the period of the simple continued fraction of argument is based on a binary quadratic relation between u n and u n+1 and properties 1+u 2 n u n+1 ∈ N and 1+u 2 n+1 un ∈ N. Due to the recurrence relation of u n , such d's are easy to be generated by hand calculation and computational mathematics via a class of explicit formulas. Besides, we consider equation x 2 − k(k
more » ... equation x 2 − k(k + 4)m 2 y 2 = −1 and show that it is solvable in integers if and only if k = 1 and m ∈ N is a divisor of 1 2 u 3n+2 for some n ∈ N ∪ {0}. The main approach for its solvability is the Fermat's method of infinite descent.
doi:10.7546/nntdm.2018.24.3.10-26 fatcat:z5v6clnkare6beiih5xghhwtpq