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Unformatted text preview: 4400/6400 PROBLEM SET 4 A sufficient number of problems: 5 for 4400 students, 8 for 6400 students. 4.1) Determine exactly which integers n , 1 ≤ n ≤ 100, are sums of two squares. 4.2) a) The first “run” of two consecutive nonnegative integers which are not sums of two squares is 6, 7. Determine the first run of at least three consecutive integers which are not sums of two squares; of at least four; of at least five. b) What is the largest k such that there exist integers n, n + 1 , . . . , n + k which are all sums of two squares. 4.3)* Prove or disprove: For any positive integer k , there exists a positive inte ger n such that none of the k +1 integers n, n +1 , . . . , n + k are sums of two squares. 4.4)* Which positive integers z can be the length of the hypotenuse of a right triangle with integer legs? (An honest triangle: each of the legs must have positive length!) Let p be an odd prime. An integer n is said to be a quadratic residue mod p if p n and n ≡ x 2 (mod p ) for some x ∈ Z . The same terminology applies to....
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 Spring '11
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 Number Theory, Integers, Prime number, zf, quadratic residue

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