For the benefit of anybody who did not know, 1. The "Pisot numbers" X are the real roots X>1 of monic integer polynomials, such that every other root r of that poly obeys |r|<1. Equivalently, the Pisot numbers are the non-integer "algebraic integers" X whose powers X^n, where n=0,1,2,3..., lie exponentially close to ordinary integers. Example: the golden ratio g=1.61803... obeying g^2=g+1 and the "plastic constant" 1.3247179572 4474602596 0908854478 0973407344 0405690173 3364534015 0503028278 5124554759 4054699347 9817872803... obeying p^3=p+1. The plastic constant is the minimum Pisot number (proven by CL Siegel in 1944). 2. The "Salem numbers" X are the real roots X>1 of monic integer polynomials, such that every other root r of that poly obeys |r|<=1, with at least one case of equality. Examples: Salem number minimal polynomial 1.72208380573905 1-z-z^2-z^3+z^4 1.40126836793985 1-z^2-z^3-z^4+z^6 1.28063815626776 1-z^3-z^4-z^5+z^8 1.17628081825992 1+z-z^3-z^4-z^5-z^6-z^7+z^9+z^10 (Lehmer) 1.21639166113827 1-z^4-z^5-z^6+z^10 1.23039143440722 1-z^3-z^5-z^7+z^10 1.20002652398739 1-z^3-z^4+z^7-z^10-z^11+z^14 Lehmer's Salem number conjecturally is minimum. This has been proven in the subcases where the polynomial is demanded to have odd coefficients or when the polynomial degree is <=54. Salem numbers S have the property that L*S^n is close to an integer for every n>=0. More precisely: for any epsilon>0, there exists an L>0, and indeed an everywhere-dense set of such L>0, such that L*S^n is within epsilon of an integer for every n>=0; and that sentence is true about Salem numbers but untrue for non-Salem non-integer reals>1. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)