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How to show that these sets are nonempty

  1. Jul 22, 2012 #1
    How to show that these sets are nonempty ([itex]\mid [/itex] means "divides")?

    Here N is an arbitrary large integer and q is some fixed integer.

    [itex]{R_{k,q}} = \{ k \in {\mathbb N}:(kN\mid k!) \wedge ((k - 1)N\mid k!) \wedge \cdots \wedge (N\mid k!) \wedge (k > Nq)\}[/itex]

    [itex]{S_{k,q}} = \{ k \in {\mathbb N}:({(2k - 1)^2}N\mid k!) \wedge ({(2k - 3)^2}N\mid k!) \wedge \ldots \wedge (N\mid k!) \wedge (k > Nq)\}[/itex]

    [itex]{T_{k,q}} = \{ k \in {\mathbb N}:({k^5}N\mid k!) \wedge ({(k - 1)^5}N\mid k!) \wedge \ldots \wedge (N\mid k!) \wedge (k > Nq)\}[/itex]

    They exist by the axiom schema of separation, but how do I determine which [itex]k[/itex] to choose so that it satisfies all the properties? Is there a general approach?
    Last edited by a moderator: Feb 4, 2013
  2. jcsd
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