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sol6 - MATH 115B SOLUTION SET VI(1 Show that there are...

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MATH 115B SOLUTION SET VI JUNE 11, 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. Solution: For an integer n , suppose that n + 1 = u 2 and ( n/ 2) + 1 = v 2 for some u and v . Then u 2 - 2 v 2 = ( n + 1) - ( n + 2) = - 1 . Now 2 = [1; 2], and the convergents C 2 k = p 2 k /q 2 k orf 2 provide solutions u = p 2 k , v = q 2 k of the equation u 2 - 2 v 2 = - 1. Thus there are infinitely many integers n for which n + 1 and ( n/ 2) + 1 are boths quares. Two such n are 48 and 1680. (2) Find the fundamental solutions of the following equations: (i) x 2 - 29 y 2 = 1; (ii) x 2 - 41 y 2 = 1. Solution: (i) Observe that 29 = [5; 2 , 1 , 1 , 2 , 10] has period 5. Hence a fundamental solution of the given equation is obtained from the convergent C 9 of 29, namely C 9 = 9801 / 1820 gives x = 9801, y = 1820. (ii) We have that 41 = [6; 2 , 2 , 12] has period 3. The convergent C 5 = 2049 / 320 gives the fundamental solution x = 2049, y = 320 of the given equation. (3)(a) Prove that whenever the equation x 2 - dy 2 = c is soluble, then it has infinitely many solutions. [Hint: If u , v satisfy x 2 - dy 2 = c and r , s satisfy x 2 - dy 2 = 1, then ( ur ± dvs ) 2 - d ( us ± vr ) 2 = (
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