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Section 3.3: Problem 9 Solution

Working problems is a crucial part of learning mathematics. No one can learn... merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.
James R. Munkres
Show that there is a representable function such that for every , , (For example, and .)
is the least such that either , or is odd, or : does not divide . Here, iff for all and , if and , then , where is the divisibility relation. This works, because iff or is odd (in which case the function is undefined, so we just return ), otherwise, is even, , and does not divide for some large , moreover, for , , and divides iff , i.e. is the least power such that does not divide .