In this paper we provide criteria for the insolvability of the Diophantine equationThis result is then used to determine the class number of the quadratic field ℚ √− .We also determine some criteria for the divisibility of the class number of the quadratic field ℚ √− and this result is then used to discuss the solvability of the Diophantine equation
For any square-free positive integer m, let H(m) be the class-number of the field Q(ζm + ζ −1 m ), where ζm is a primitive m-th root of unity. We show that if m = {3(8g + 5)} 2 − 2 is a square-free integer, where g is a positive integer, then H(4m) > 1. Similar result holds for a square-free integer m = {3(8g +7)} 2 −2, where g is a positive integer. We also show that n|H(4m) for certain positive integers m and n. Mathematics Subject Classification (2010): 11R29, 11R18, 11R11.
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