Unformatted text preview: integers. 4. Let α, β be positive irrational numbers with 1 /α + 1 /β = 1. Show that the sequences f ( n ) = b αn c and g ( n ) = b βn c , n = 1 , 2 , 3 , . . . , are disjoint and their union is N . 5. Prove that the sum of the squares of ﬁve consecutive integers is never a perfect square. 6. Show that there are inﬁnitely many triples ( x, y, z ) of positive integers such that x 2 + y 2 + z 2 = xyz . 1...
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 Spring '08
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 Number Theory, Integers, Prime Numbers, Natural number, Prime number, positive integers, squarefree positive integers

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