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 (1902–1955), 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 nth power of a Pisot number approaches integers as n approaches infinity. For example, consider powers of φ, such as φ²¹ = 24476.0000409. 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 nth 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 1 Table of Pisot numbers 2 See also 3 External links 4 References

[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

• Salem number

[edit] External links

• Pisot number, Encyclopedia of Mathematics
• Terr, David and Weisstein, Eric W., Pisot Number at MathWorld.

[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