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
with a and b both integers, or in other cases both half an odd integer, we get
The conditions are then
- α > 1
and
- − 1 < α' < 1.
This condition is satisfied by the golden ratio φ. We have
and
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 November 1902 - 20 April 1955), 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 φ, such as φ21 = 24476.000409. 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 lowest Pisot-Vijayaraghavan number is the unique real solution of x3 − x − 1, known as the plastic number (approximatively 1.324718).
The lowest accumulation point of the set of Pisot-Vijayaraghavan numbers is the golden ratio 
. The set of all Pisot-Vijayaraghavan numbers is nowhere dense because it is a closed and countable set.
Contents |
[edit] Table of Pisot numbers
Here are the 38 Pisot numbers less than 1.618, in increasing order.
| Value | Root of... | |
|---|---|---|
| 1 | 1.3247179572447460260 | x3 − x − 1 |
| 2 | 1.3802775690976141157 | x4 − x3 − 1 |
| 3 | 1.4432687912703731076 | x5 − x4 − x3 + x2 − 1 |
| 4 | 1.4655712318767680267 | x3 − x2 − 1 |
| 5 | 1.5015948035390873664 | x6 − x5 − x4 + x2 − 1 |
| 6 | 1.5341577449142669154 | x5 − x3 − x2 − x − 1 |
| 7 | 1.5452156497327552432 | x7 − x6 − x5 + x2 − 1 |
| 8 | 1.5617520677202972947 | x6 − 2x5 + x4 − x2 + x − 1 |
| 9 | 1.5701473121960543629 | x5 − x4 − x2 − 1 |
| 10 | 1.5736789683935169887 | x8 − x7 − x6 + x2 − 1 |
| 11 | 1.5900053739013639252 | x7 − x5 − x4 − x3 − x2 − x − 1 |
| 12 | 1.5911843056671025063 | x9 − x8 − x7 + x2 − 1 |
| 13 | 1.6013473337876367242 | x7 − x6 − x4 − x2 − 1 |
| 14 | 1.6017558616969832557 | x10 − x9 − x8 + x2 − 1 |
| 15 | 1.6079827279282011499 | x9 − x7 − x6 − x5 − x4 − x3 − x2 − x − 1 |
| 16 | 1.6081283851873869594 | x11 − x10 − x9 + x2 − 1 |
| 17 | 1.6119303965641198198 | x9 − x8 − x6 − x4 − x2 − 1 |
| 18 | 1.6119834212464921559 | x12 − x11 − x10 + x2 − 1 |
| 19 | 1.6143068232571485146 | x11 − x9 − x8 − x7 − x6 − x5 − x4 − x3 − x2 − x − 1 |
| 20 | 1.6143264149391271041 | x13 − x12 − x11 + x2 − 1 |
| 21 | 1.6157492027552106107 | x11 − x10 − x8 − x6 − x4 − x2 − 1 |
| 22 | 1.6157565175408433755 | x14 − x13 − x12 + x2 − 1 |
| 23 | 1.6166296843945727036 | x13 − x11 − x10 − x9 − x8 − x7 − x6 − x5 − x4 − x3 − x2 − x − 1 |
| 24 | 1.6166324353879050082 | x15 − x14 − x13 + x2 − 1 |
| 25 | 1.6171692963550925635 | x13 − x12 − x10 − x8 − x6 − x4 − x2 − 1 |
| 26 | 1.6171703361720168476 | x16 − x15 − x14 + x2 − 1 |
| 27 | 1.6175009054313240144 | x15 − x13 − x12 − x11 − x10 − x9 − x8 − x7 − x6 − x5 − x4 − x3 − x2 − x − 1 |
| 28 | 1.6175012998129095573 | x17 − x16 − x15 + x2 − 1 |
| 29 | 1.6177050699575566445 | x15 − x14 − x12 − x10 − x8 − x6 − x4 − x2 − 1 |
| 30 | 1.6177052198884550971 | x18 − x17 − x16 + x2 − 1 |
| 31 | 1.6178309287889738637 | x17 − x15 − x14 − x13 − x12 − x11 − x10 − x9 − x8 − x7 − x6 − x5 − x4 − x3 − x2 − x − 1 |
| 32 | 1.6178309858778122988 | x19 − x18 − x17 + x2 − 1 |
| 33 | 1.6179085817671650120 | x17 − x16 − x14 − x12 − x10 − x8 − x6 − x4 − x2 − 1 |
| 34 | 1.6179086035278053858 | x20 − x19 − x18 + x2 − 1 |
| 35 | 1.6179565199535642392 | x19 − x17 − x16 − x15 − x14 − x13 − x12 − x11 − x10 − x9 − x8 − x7 − x6 − x5 − x4 − x3 − x2 − x − 1 |
| 36 | 1.6179565282539765702 | x21 − x20 − x19 + x2 − 1 |
| 37 | 1.6179861253852491516 | x19 − x18 − x16 − x14 − x12 − x10 − x8 − x6 − x4 − x2 − 1 |
| 38 | 1.6179861285528618287 | x22 − x21 − x20 + x2 − 1 |
The number 
is a PV number that is not a unit, since it satisfies the equation x2-4x+2=0.
Every real algebraic number field contains a PV number that generates this field. In quadratic and cubic fields it is not hard to find a unit that is a PV number.
[edit] See also
[edit] External links
[edit] 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
