Pisot-Vijayaraghavan number

From Wikipedia, the free encyclopedia

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 or silver number (approximatively 1.324718).

The lowest accumulation point of the set of Pisot-Vijayaraghavan numbers is the golden ratio .

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

[edit] See also

• Salem number

[edit] External links

• Pisot number, Encyclopedia of Mathematics [1]
• Pisot number, Mathworld [2]

[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