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NT2009HW5

NT2009HW5 - 4400/6400 PROBLEM SET 4 A sufficient number of...

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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 non-negative 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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NT2009HW5 - 4400/6400 PROBLEM SET 4 A sufficient number of...

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