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# Solving Problems From The Back - 2

Claim: The only solutions are $$f(x) = 1$$ and $$f(x) = -1$$.
Exercise 2: Show that these functions satisfy the conditions.

Now, what possible result could lead to this conclusion?

Breadcrumb 1: We want to show that $$f( 2^n) = 1$$ or $$-1$$ for all integers $$n$$.
Exercise 3: Show that if Breadcrumb 1 is true, then the claim is true.

Breadcrumb 2: We want to show that for any integer $$n$$, there exists an integer $$k$$ such that $$f( 2^n) \mid f ( k)$$ and $$f(2^n) \mid f(2^k)$$.
Exercise 4: Show that if Breadcrumb 2 is true, then Breadcrumb 1 is true.

Ponder this, and then move on to the next note in this set.

Note by Calvin Lin
3 years ago