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The Pell numbers Pn and qn are defined byPo= 0qo = 1P1 = 1qi= 1Pn = 2Pn-l + Pn-2qn =

ISBN: 9780073383149 413

Solution for problem 15 Chapter 15

Elementary Number Theory | 7th Edition

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Problem 15

The Pell numbers Pn and qn are defined byPo= 0qo = 1P1 = 1qi= 1Pn = 2Pn-l + Pn-2qn = 2qn-l + qn-2This gives us the two sequences0, 1,2,5, 12,29, 70, 169,408, . . .1, 1,3, 7, 17,41,99,239,577, . . .n :'.'.: 2n :'.'.: 2If a = 1 + ,J2 and fJ = 1 - ,J2, show that the Pell numbers can be expressed asan - {JnPn =2for n :'.'.: 0.[Hint: Mimic the argument on page 296, noting that a and fJ are roots of the equationx2 - 2x - 1 = O.]

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