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Pisot Number

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A Pisot number is a positive algebraic integer greater than 1 all of whose conjugate elements have absolute value less than 1. A real quadratic algebraic integer greater than 1 and of degree 2 or 3 is a Pisot number if its norm is equal to +/-1. The golden ratio phi (denoted theta_0 when considered as a Pisot number) is an example of a Pisot number since it has degree two and norm -1.

The smallest Pisot number is given by the positive root theta_1=1.324717957... (OEIS A060006) of

x^3-x-1=0,
(1)

known as the plastic constant. This number was identified as the smallest known by Salem (1944), and proved to be the smallest possible by Siegel (1944).

PisotConstant

Pisot constants give rise to almost integers. For example, the larger the power to which theta_1 is taken, the closer theta_1^n-|_theta_1^n_|, where |_x_| is the floor function, is to either 0 or 1 (Trott 2004). For example, the spectacular example theta_1^(27369) is within 1.18463×10^(-1671) of an integer (Trott 2004, pp. 8-9).

The powers of theta_1 for which this quantity is closer to 0 are 1, 3, 4, 5, 6, 7, 8, 11, 12, 14, 17, ... (OEIS A051016), while those for which it is closer to 1 are 2, 9, 10, 13, 15, 16, 18, 20, 21, 23, ... (OEIS A051017).

Siegel also identified the second smallest Pisot numbers as the positive root theta_2=1.38027756... (OEIS A086106) of

x^4-x^3-1=0,
(2)

showed that theta_1 and theta_2 are isolated, and showed that the positive roots of each polynomial

x^n(x^2-x-1)+x^2-1
(3)

for n=1, 2, 3, ...,

x^n-(x^(n+1)-1)/(x^2-1)
(4)

for n=3, 5, 7, ..., and

x^n-(x^(n-1)-1)/(x-1)
(5)

for n=3, 5, 7, ... are Pisot numbers.

All the Pisot numbers less than the golden ratio phi are known (Dufresnoy and Pisot 1955). Some small Pisot numbers and their minimal polynomials are given in the following table. The latter two entries are from Boyd (1977).

Pisot numberconstantOEISvalueorderpolynomial
theta_0golden ratio phiA0016221.6180339887...2x^2-x-1
theta_1plastic constant PA0600061.3247179572...3x^3-x-1
theta_2chiA0861061.3802775690...4x^4-x^3-1
theta_3A2287771.4432687912...5x^5-x^4-x^3+x^2-1
theta_4supergolden ratio psiA0925261.4655712318...3x^3-x^2-1
theta_5A2935081.5015948035...6x^6-x^5-x^4+x^2-1
theta_6A2935091.5341577449...5x^5-x^3-x^2-x-1
theta_7A2935571.5452156497...7x^7-x^6-x^5+x^2-1
theta_8A3740021.5617520677...6x^6-2x^5+x^4-x^2+x-1
theta_9A2935061.5701473121...5x^5-x^4-x^2-1
theta_(10)A3740031.5736789683...8x^8-x^7-x^6+x^2-1
theta_(16)1.6216584885...16x^(16)-2x^(15)+2x^(14)-3x^(13)+2x^(12)-2x^(11)+x^(10)+x^7-x^6+2x^5-2x^4+2x^3-2x^2+x-1
theta_(20)1.8374664495...20x^(20)-2x^(19)+x^(17)-x^(16)+x^(14)-x^(13)+x^(11)-x^9+x^7-x^6+x^4-x^3+x-1

As can be seen in the preceding table, the orders of the minimal polynomials for the smallest, second smallest, etc. Pisot numbers are given by 3, 4, 5, 3, 6, 5, 7, 6, 5, 8, ... (OEIS A381191).

Pisot numbers originally arose in the consideration of

frac(x)=x-|_x_|
(6)

where frac(x) denotes the fractional part of x and |_x_| is the floor function. Letting theta be a number greater than 1 and lambda a positive number, for a given lambda, the sequence of numbers frac(lambdatheta^n) for n=1, 2, ... is an equidistributed sequence in the interval (0, 1) when theta does not belong to a lambda-dependent exceptional set S of measure zero (Koksma 1935). Pisot (1938) and Vijayaraghavan (1941) independently studied the exceptional values of theta, and Salem (1943) proposed calling such values Pisot-Vijayaraghavan numbers.

Pisot (1938) subsequently proved the fact that if theta is chosen such that there exists a lambda!=0 for which the series

sum_(n=0)^inftysin^2(pilambdatheta^n)
(7)

converges, then theta is an algebraic integer whose conjugates all (except for itself) have modulus <1, and lambda is an algebraic integer of the field K(theta). Vijayaraghavan (1940) proved that the set of Pisot numbers has infinitely many limit points. Salem (1944) proved that the set of Pisot numbers is closed. The proof of this theorem is based on the lemma that for a Pisot number theta, there always exists a number lambda such that 1<=lambda<theta and the following inequality is satisfied:

sum_(n=0)^inftysin^2(pilambdatheta^n)<=(pi^2(2theta+1)^2)/((theta-1)^2).
(8)

See also

Almost Integer, Equidistributed Sequence, Golden Ratio, Lehmer's Mahler Measure Problem, Plastic Constant, Salem Constants, Supergolden Ratio, Weyl's Criterion

Portions of this entry contributed by David Terr

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References

Bell, J. P. and Hare, K. G. "Properties of Z(q) for q a Pisot number." http://www.math.uwaterloo.ca/~kghare/Preprints/PDF/P17_Zq.pdf.Bertin, M. J. and Pathiaux-Delefosse, A. Conjecture de Lehmer et petits nombres de Salem. Kingston: Queen's Papers in Pure and Applied Mathematics, 1989.Bertin, M. J.; Decomps-Guilloux, A.; Grandet-Hugot, M.; Pathiaux-Delefosse, M.; and Schreiber, J. P. Pisot and Salem Numbers. Basel: Birkhäuser, 1992.Borwein, P. and Hare, K. G. "Some Computations on Pisot and Salem Numbers." CECM-00:148, 18 May. http://www.cecm.sfu.ca/preprints/2000pp.html#00:148.Boyd, D. W. "Small Salem Numbers." Duke Math. J. 44, 315-328, 1977.Boyd, D. W. "Pisot and Salem Numbers in Intervals of the Real Line." Math. Comput. 32, 1244-1260, 1978.Boyd, D. W. "Pisot Numbers in the Neighbourhood of a Limit Point. II." Math. Comput. 43, 593-602, 1984.Boyd, D. W. "Pisot Numbers in the Neighbourhood of a Limit Point. I." J. Number Theory 21, 17-43, 1985.Dubickas, A. "A Note on Powers of Pisot Numbers." Publ. Math. Debrecen 56, 141-144, 2000.Dufresnoy, J. and Pisot, C. "Étude de certaines fonctions méromorphes bornées sur le cercle unité, application à un ensemble fermé d'entiers algébriques." Ann. Sci. École Norm. Sup. 72, 69-92, 1955.Erdős, P.; Joó, M.; and Schnitzer, F. J. "On Pisot Numbers." Ann. Univ. Sci. Budapest, Eőtvős Sect. Math. 39, 95-99, 1997.Katai, I. and Kovacs, B. "Multiplicative Functions with Nearly Integer Values." Acta Sci. Math. 48, 221-225, 1985.Koksma, J. F. "Ein mengentheoretischer Satz über die Gleichverteilung modulo Eins." Comp. Math. 2, 250-258, 1935.Le Lionnais, F. Les nombres remarquables. Paris: Hermann, pp. 38 and 148, 1983.Luca, F. "On a Question of G. Kuba." Arch. Math. (Basel) 74, 269-275, 2000.Pegg, E. Jr. "Shattering the Plane with Twelve New Substitution Tilings Using 2, phi, psi, chi, rho." Mar. 7, 2019. https://blog.wolfram.com/2019/03/07/shattering-the-plane-with-twelve-new-substitution-tilings-using-2-phi-psi-chi-rho/.Pisot, C. "La répartition modulo 1 et les nombres algébriques." Annali di Pisa 7, 205-248, 1938.Salem, R. "Sets of Uniqueness and Sets of Multiplicity." Trans. Amer. Math. Soc. 54, 218-228, 1943.Salem, R. "A Remarkable Class of Algebraic Numbers. Proof of a Conjecture of Vijayaraghavan." Duke Math. J. 11, 103-108, 1944.Salem, R. "Power Series with Integral Coefficients." Duke Math. J. 12, 153-172, 1945.Siegel, C. L. "Algebraic Numbers whose Conjugates Lie in the Unit Circle." Duke Math. J. 11, 597-602, 1944.Sloane, N. J. A. Sequences A001622/M4046, A051016, A051017, A060006, A086106, A092526, A228777, A293506, A293508, A293509, A293557, A374002, A374003, and A381191 in "The On-Line Encyclopedia of Integer Sequences."Trott, M. The Mathematica GuideBook for Programming. New York: Springer-Verlag, 2004. http://www.mathematicaguidebooks.org/.Vijayaraghavan, T. "On the Fractional Parts of the Powers of a Number, II." Proc. Cambridge Phil. Soc. 37, 349-357, 1941.

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Pisot Number

Cite this as:

Terr, David and Weisstein, Eric W. "Pisot Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PisotNumber.html

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