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**Unformatted text preview: **MATH 115B PROBLEM SET VI
JUNE 5, 2007
(1) Show that there are inﬁnitely many even integers n with the property that both n + 1
and (n/2) + 1 are perfect squares. Exhibit two such integers.
(2) Find the fundamental solutions of the following equations:
(i) x2 − 29y 2 = 1;
(ii) x2 − 41y 2 = 1.
(3)(a) Prove that whenever the equation x2 − dy 2 = c is soluble, then it has inﬁnitely
many solutions.
[Hint: If u, v satisfy x2 − dy 2 = c and r, s satisfy x2 − dy 2 = 1, then
(ur ± dvs)2 − d(us ± vr)2 = (u2 − dv 2 )(r2 − ds2 ) = c.]
(b) Given that x = 16, y = 6 is a solution of x2 − 7y 2 = 4, ﬁnd two other positive solutions. 1 ...

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