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Number λ ≈ 1.17628....Aren't You Special?

We just celebrated the special number Pi. What is it that makes a number special? Think about Pi, e, Phi, even i = Sqrt(-1). These numbers are special because of their special properties.

But, who decides what properties of a number are special enough to elevate the number to its own special status?

For example, λ ≈ 1.17628 is special. It is called Lehmer's number. Its special properties?

  • It is the largest real root of the polynomial x10 + x9 - x7 - x6 - x5 - x4 - x3 + x + 1...does that get you excited?
  • It is conjectured to be the smallest size of an algebraic integer α larger than 1 (i.e. size means the "Mahler measure of the minimal polynomial of α")...are you excited yet?
  • It is the only one of its algebraic conjugates that lies outside the unit circle...making it either a Salem number or a Pisot number...bet that excites you?
  • It is the special geometric dilatation of a monodromy φ of the (-2,3,7)-pretzel knot...Food...now you are excited!
I will stop here. My point, in case it was missed, is that a number's specialness is relative to the audience's understanding of its special properties. Pi is special to almost anyone, because some of its special properties are understandable and deemed special by so many. But, like the Lehmer number, one could state extremely-estoteric properties of pi that would elevate its standing to specialness only amongst certain mathematicians.

Source: E. Hironaka's "What is Lehmer's Number?" Notices of the AMS, March 2009, pp. 374-375.