Pisot Number

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

The smallest Pisot number is given by the positive root (OEIS A060006) of

(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).

Pisot constants give rise to almost integers. For example, the distance of from the nearest integer tends to 0 as increases. Thus the fractional part is arbitrarily close to either 0 or 1 for sufficiently large , where is the floor function (Trott 2004). For example, the spectacular example is within of an integer (Trott 2004, pp. 8-9).

The powers of 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 number, the superplastic ratio, as the positive root (OEIS A086106) of

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

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for , 2, 3, ...,

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for , 5, 7, ..., and

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for , 5, 7, ... are Pisot numbers.

All the Pisot numbers less than the golden ratio 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 number	constant	OEIS	value	order	polynomial
	golden ratio	A001622	1.6180339887...	2
	plastic constant	A060006	1.3247179572...	3
	superplastic ratio	A086106	1.3802775690...	4
		A228777	1.4432687912...	5
	supergolden ratio	A092526	1.4655712318...	3
		A293508	1.5015948035...	6
		A293509	1.5341577449...	5
		A293557	1.5452156497...	7
		A374002	1.5617520677...	6
		A293506	1.5701473121...	5
		A374003	1.5736789683...	8
			1.6216584885...	16
			1.8374664495...	20

Pisot numbers occur among the scale factors in an infinite family of triangle dissections of an equilateral triangle into similar triangles. As detailed there, the squares of the scale factors in the first three even-indexed members are the tribonacci constant, , and ; later members do not give Pisot numbers. The first member also has a scale factor .

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

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

Pisot (1938, p. 233, Theorem III) proved that if and are real numbers for which the series

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converges, then is an algebraic integer whose conjugates all (except for itself) have modulus , and belongs to the field . 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 , there always exists a number such that and the following inequality is satisfied:

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