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Salem number
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== Relation with Pisot–Vijayaraghavan numbers== The smallest known Salem number is the largest real root of '''Lehmer's polynomial'''<!-- redirects here --> (named after [[Derrick Henry Lehmer]]) :<math>P(x) = x^{10} + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1,</math> which is about <math>x = 1.17628</math>: it is [[conjecture]]d that it is indeed the smallest Salem number, and the smallest possible [[Mahler measure]] of an [[irreducible polynomial|irreducible]] non-[[cyclotomic polynomial|cyclotomic]] polynomial.<ref>Borwein (2002) p.16</ref> Lehmer's polynomial is a factor of the shorter [[degree of a polynomial|degree]]-12 polynomial, :<math>Q(x) = x^{12} - x^7 - x^6 - x^5 + 1,</math> all twelve roots of which satisfy the relation<ref>D. Bailey and D. Broadhurst, [http://crd-legacy.lbl.gov/~dhbailey/dhbpapers/ladder.pdf A Seventeenth Order Polylogarithm Ladder]</ref> :<math>x^{630}-1 = \frac{(x^{315}-1)(x^{210}-1)(x^{126}-1)^2(x^{90}-1)(x^{3}-1)^3(x^{2}-1)^5(x-1)^3 }{(x^{35}-1)(x^{15}-1)^2(x^{14}-1)^2(x^{5}-1)^6\,x^{68}}</math> Salem numbers can be constructed from [[Pisot–Vijayaraghavan number]]s. To recall, the smallest of the latter is the unique real root of the [[cubic polynomial]], :<math>x^3 - x - 1,</math> known as the ''[[plastic ratio]]'' and approximately equal to 1.324718. This can be used to generate a family of Salem numbers including the smallest one found so far. The general approach is to take the minimal polynomial <math>P(x)</math> of a Pisot–Vijayaraghavan number and its [[reciprocal polynomial]], <math>P^*(x)</math>, and solve the equation, :<math>x^n P(x) = \pm P^{*}(x)</math> for [[integer]] <math>n</math> above a bound. Subtracting one side from the other, factoring, and disregarding trivial factors will then yield the minimal polynomial of certain Salem numbers. For example, using the negative case of the above, :<math>x^n(x^3 - x - 1) = -(x^3 + x^2 - 1)</math> then for <math>n=8</math>, this factors as, :<math>(x-1)(x^{10} + x^9 -x^7 -x^6 -x^5 -x^4 -x^3 +x +1) = 0</math> where the [[Decic equation|decic]] is Lehmer's polynomial. Using higher <math>n</math> will yield a family with a root approaching the plastic ratio. This can be better understood by taking <math>n</math>th roots of both sides, :<math>x(x^3 - x - 1)^{1/n} = \pm (x^3 + x^2 - 1)^{1/n} </math> so as <math>n</math> goes higher, <math>x</math> will approach the solution of <math>x^3 - x - 1=0</math>. If the positive case is used, then <math>x</math> approaches the plastic ratio from the opposite direction. Using the minimal polynomial of the next smallest Pisot–Vijayaraghavan number gives :<math>x^n(x^4 - x^3 - 1) = -(x^4 + x - 1),</math> which for <math>n=7</math> factors as :<math>(x-1)(x^{10} - x^6 - x^5 - x^4 + 1) = 0,</math> a decic not generated in the previous and has the root <math>x=1.216391\ldots</math> which is the 5th smallest known Salem number. As <math>n\to\infty</math>, this family in turn tends towards the larger real root of <math>x^4 - x^3 - 1 = 0</math>.
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