What are some things we can prove they must exist, but have no idea what they are?

What are some things we can prove they must exist, but have no idea what they are?

Examples I can think of:

  • Values of the Busy beaver function: It is a well-defined function, but not computable.
  • It had been long known that there must be self-replicating patterns in Conway’s Game of Life, but the first such pattern has only been discovered in 2010
  • By the Well-ordering theorem under the assumptions of the axiom of choice, every set admits a well-ordering. (See also Is there a known well ordering of the reals?)

Answer

Not sure if this satisfies the requirement that we “have no idea what they are”, but the extremely strange Mill’s constant seems worth mentioning here: There is supposed to be some real number r>0 with the property that the integer part of r3n
is prime for every natural n. It is not known if r is rational and as far as I know not even a numerical approximation of r is possible without assuming the Riemann hypothesis.

Source: Wikipedia

Attribution
Source : Link , Question Author : Loreno Heer , Answer Author : Ooker

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