prob6 - MATH 115B PROBLEM SET VI JUNE 5, 2007 (1) Show that...

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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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This document was uploaded on 12/26/2011.

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