JP Journal of Algebra, Number Theory and Applications
Volume 4, Issue 2, Pages 353 - 362
(August 2004)
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INFINITELY MANY QUADRATIC DIOPHANTINE EQUATIONS SOLVABLE
EVERYWHERE LOCALLY, BUT NOT SOLVABLE GLOBALLY
R. A. Mollin (Canada)
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Abstract: We present an infinite
class of integers 2c, which turn out to
be Richaud-Degert type radicands, for which 2x2
– cy2 = – 1 has no integer solutions, but for which
2x2
– cy2 º
– 1(mod n) has integer solutions for any
n
Î
N.
This explains a topic often seen in introductory
number theory courses, that appears as a
curiosity, yet for which we are able to give the
underlying reasons in terms of simple continued
fraction expansions. |
Keywords and phrases: quadratic Diophantine
equations, simple continued fractions, congruences. |
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