4802NumTheory07

4802NumTheory07 - integers 4 Let α β be positive...

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Math 4802 Fall 2007 Number Theory Problems 1. a. Find all positive integers n for which 7 divides 2 n - 1. b. Show that there is no positive integer n for which 7 divides 2 n + 1. 2. Find all prime numbers p whose base 10 expansion consists of alternating 1’s and 0’s, beginning and ending with 1. 3. A positive integer n is square-free if it is not divisible by p 2 for any prime number p . Assuming the known fact that X p prime 1 p 2 < 0 . 46 , show that every integer n 2 is the sum of two square-free positive
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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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