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Pisot–Vijayaraghavan number
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==Quadratic irrationals== If <math>\alpha\,</math> is a [[quadratic irrational]] there is only one other conjugate, <math>\alpha'</math>, obtained by changing the sign of the [[square root]] in <math>\alpha</math> from :<math>\alpha = a + \sqrt D \text{ to } \alpha' = a - \sqrt D\, </math> or from : <math>\alpha = \frac{a + \sqrt D}{2}\text{ to }\alpha' = \frac{a - \sqrt D}{2}.\,</math> Here ''a'' and ''D'' are integers and in the second case ''a'' is [[parity (mathematics)|odd]] and ''D'' is [[modular arithmetic|congruent]] to 1 modulo 4. The required conditions are ''α'' > 1 and −1 < ''α''' < 1. These are satisfied in the first case exactly when ''a'' > 0 and either <math>(a-1)^2 < D < a^2</math> or <math>a^2 < D < (a+1)^2</math>, and are satisfied in the second case exactly when <math>a>0</math> and either <math>(a-2)^2 < D < a^2</math> or <math>a^2 < D < (a+2)^2</math>. Thus, the first few quadratic irrationals that are PV numbers are: <div class="overflowbugx" style="overflow-x:auto;"> {| class="wikitable" |- ! Value !! Root of... !! Numerical value |- style="height:50px" | <math>\frac{1+\sqrt{5}}{2}</math> ||<math>x^2-x-1</math> || 1.618033... {{OEIS2C|A001622}} (the [[golden ratio]]) |- style="height:50px" | <math>1+\sqrt{2}\,</math> ||<math>x^2-2x-1</math> || 2.414213... {{OEIS2C|A014176}} (the [[silver ratio]]) |- style="height:50px" | <math>\frac{3+\sqrt{5}}{2}</math> ||<math>x^2-3x+1</math> || 2.618033... {{OEIS2C|A104457}} (the golden ratio squared) |- style="height:50px" | <math>1+\sqrt{3}\,</math> ||<math>x^2-2x-2</math> || 2.732050... {{OEIS2C|A090388}} |- style="height:50px" | <math>\frac{3+\sqrt{13}}{2}</math> ||<math>x^2-3x-1</math> || 3.302775... {{OEIS2C|A098316}} (the third [[metallic mean]]) |- style="height:50px" | <math>2+\sqrt{2}\,</math> ||<math>x^2-4x+2</math> || 3.414213... |- style="height:50px" | <math>\frac{3+\sqrt{17}}{2}</math> ||<math>x^2-3x-2</math> || 3.561552.. {{OEIS2C|A178255}}. |- style="height:50px" | <math>2+\sqrt{3}\,</math> ||<math>x^2-4x+1</math> || 3.732050... {{OEIS2C|A019973}} |- style="height:50px" | <math>\frac{3+\sqrt{21}}{2}</math> ||<math>x^2-3x-3</math> || 3.791287...{{OEIS2C|A090458}} |- style="height:50px" | <math>2+\sqrt{5}\,</math> ||<math>x^2-4x-1</math> || 4.236067... {{OEIS2C|A098317}} (the fourth metallic mean) |} </div>
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