1993 UNCO Math Contest II Problems/Problem 6
(a) Find integers and so that
(b) Conjecture a general rule that is being illustrated here.
(c) Prove your conjecture.
(a) We can rewrite the given equation as . Use difference of squares to obtain . Since is a prime we conclude that , giving us .
(b) It is not too hard to notice that the LHS above is and the RHS above is for . We will prove that the LHS RHS for all integers (although the proof extends to real numbers) in (c).
(c) We expand the LHS to obtain Thus LHS = RHS and we are done. ~AK2006
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