Pisot-Vijayaraghavan number
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In mathematics, a Pisot-Vijayaraghavan number, also called simply a Pisot number or a PV number, is an algebraic integer α which is real and exceeds 1, but such that its conjugate elements are all less than 1 in absolute value.
For example, if α is a quadratic irrational there is only one other conjugate: α′, obtained by changing the sign of the square root in α; from
- <math>\alpha = a + b \sqrt d</math>
with a and b both integers, or in other cases both half an odd integer, we get
- <math>\alpha' = a - b \sqrt d</math>
The conditions are then
- <math>\alpha > 1 \textrm{\ and } -1 < \alpha'< 1.</math>
This condition is satisfied by the golden ratio Φ. We have
- <math>\Phi = \frac{1 + \sqrt 5}{2} > 1</math>
and
- <math>\Phi' = \frac{1 - \sqrt 5} 2 = \frac{-1}\Phi .</math>
The general condition was investigated by G. H. Hardy in relation with a problem of diophantine approximation. This work was followed up by Tirukkannapuram Vijayaraghavan (30 November1902 - 20 April1955), an Indian mathematician from the Madras region who came to Oxford to work with Hardy in the mid-1920s. The same condition also occurs in some problems on Fourier series, and was later investigated by Charles Pisot. The name now commonly used comes from both of those authors.
Pisot-Vijayaraghavan numbers can be used to generate almost integers: the n-th power of a Pisot number "approaches integers" as n tends to infinity. For example, consider powers of <math>\Phi</math>, such as <math>\Phi^{21} = 24476.000409</math>. The effect can be even more pronounced for Pisot-Vijayaraghavan numbers generated from equations of higher degree.
This property stems from the fact that for each n, the sum of n-th powers of an algebraic integer x and its conjugates is exactly an integer; when x is a Pisot number, the n-th powers of the (other) conjugates tend to 0 as n tends to infinity. The converse holds: if x is a real number > 1 and there is a sequence of integers <math>a_n</math> so that <math>\lim_{n \to \infty } x^n - a_n = 0</math>, then x is a Pisot-Vijayaraghavan number.
The lowest Pisot-Vijayaraghavan number is the unique real solution of <math>x^3 - x - 1</math>, known as the plastic number or silver number (approximatively 1.324718).
The lowest accumulation point of the set of Pisot-Vijayaraghavan numbers is the golden ratio <math>\Phi = \frac{1 + \sqrt 5}{2} \approx 1.618033</math>.
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Table of Pisot numbers
Here are the 38 Pisot numbers less than 1.618, in order of size.
1: 1.3247179572447460260 <math>x^3-x-1</math>
2: 1.3802775690976141157 <math>x^4-x^3-1</math>
3: 1.4432687912703731076 <math>x^5-x^4-x^3+x^2-1</math>
4: 1.4655712318767680267 <math>x^3-x^2-1</math>
5: 1.5015948035390873664 <math>x^6-x^5-x^4+x^2-1</math>
6: 1.5341577449142669154 <math>x^5-x^3-x^2-x-1</math>
7: 1.5452156497327552432 <math>x^7-x^6-x^5+x^2-1</math>
8: 1.5617520677202972947 <math>x^6-2x^5+x^4-x^2+x-1</math>
9: 1.5701473121960543629 <math>x^5-x^4-x^2-1</math>
10: 1.5736789683935169887 <math>x^8-x^7-x^6+x^2-1</math>
11: 1.5900053739013639252 <math>x^7-x^5-x^4-x^3-x^2-x-1</math>
12: 1.5911843056671025063 <math>x^9-x^8-x^7+x^2-1</math>
13: 1.6013473337876367242 <math>x^7-x^6-x^4-x^2-1</math>
14: 1.6017558616969832557 <math>x^{10}-x^9-x^8+x^2-1</math>
15: 1.6079827279282011499 <math>x^9-x^7-x^6-x^5-x^4-x^3-x^2-x-1</math>
16: 1.6081283851873869594 <math>x^{11}-x^{10}-x^9+x^2-1</math>
17: 1.6119303965641198198 <math>x^9-x^8-x^6-x^4-x^2-1</math>
18: 1.6119834212464921559 <math>x^{12}-x^{11}-x^{10}+x^2-1</math>
19: 1.6143068232571485146 <math>x^{11}-x^9-x^8-x^7-x^6-x^5-x^4-x^3-x^2-x-1</math>
20: 1.6143264149391271041 <math>x^{13}-x^{12}-x^{11}+x^2-1</math>
21: 1.6157492027552106107 <math>x^{11}-x^{10}-x^8-x^6-x^4-x^2-1</math>
22: 1.6157565175408433755 <math>x^{14}-x^{13}-x^{12}+x^2-1</math>
23: 1.6166296843945727036 <math>x^{13}-x^{11}-x^{10}-x^9-x^8-x^7-x^6-x^5-x^4-x^3-x^2-x-1</math>
24: 1.6166324353879050082 <math>x^{15}-x^{14}-x^{13}+x^2-1</math>
25: 1.6171692963550925635 <math>x^{13}-x^{12}-x^{10}-x^8-x^6-x^4-x^2-1</math>
26: 1.6171703361720168476 <math>x^{16}-x^{15}-x^{14}+x^2-1</math>
27: 1.6175009054313240144 <math>x^{15}-x^{13}-x^{12}-x^{11}-x^{10}-x^9-x^8-x^7-x^6-x^5-x^4-x^3-x^2-x-1</math>
28: 1.6175012998129095573 <math>x^{17}-x^{16}-x^{15}+x^2-1</math>
29: 1.6177050699575566445 <math>x^{15}-x^{14}-x^{12}-x^{10}-x^8-x^6-x^4-x^2-1</math>
30: 1.6177052198884550971 <math>x^{18}-x^{17}-x^{16}+x^2-1</math>
31: 1.6178309287889738637 <math>x^{17}-x^{15}-x^{14}-x^{13}-x^{12}-x^{11}-x^{10}-x^9-x^8-x^7-x^6-x^5-x^4-x^3-x^2-x-1</math>
32: 1.6178309858778122988 <math>x^{19}-x^{18}-x^{17}+x^2-1</math>
33: 1.6179085817671650120 <math>x^{17}-x^{16}-x^{14}-x^{12}-x^{10}-x^8-x^6-x^4-x^2-1</math>
34: 1.6179086035278053858 <math>x^{20}-x^{19}-x^{18}+x^2-1</math>
35: 1.6179565199535642392 <math>x^{19}-x^{17}-x^{16}-x^{15}-x^{14}-x^{13}-x^{12}-x^{11}-x^{10}-x^9-x^8-x^7-x^6-x^5-x^4-x^3-x^2-x-1</math>
36: 1.6179565282539765702 <math>x^{21}-x^{20}-x^{19}+x^2-1</math>
37: 1.6179861253852491516 <math>x^{19}-x^{18}-x^{16}-x^{14}-x^{12}-x^{10}-x^8-x^6-x^4-x^2-1</math>
38: 1.6179861285528618287 <math>x^{22}-x^{21}-x^{20}+x^2-1</math>
See also
External links
Pisot number, Encyclopedia of Mathematics http://eom.springer.de/p/p120130.htm
Pisot number, Mathworld http://mathworld.wolfram.com/PisotNumber.html
References
M.J. Bertin, A. Decomps-Guilloux, M. Grandet-Hugot, M. Pathiaux-Delefosse, J.P. Schreiber, "Pisot and Salem Numbers" , Birkhäuser (1992)
D.W. Boyd, "Pisot and Salem numbers in intervals of the real line" Math. Comp. , 32 (1978) pp. 1244–1260