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 Sums of Squares (Posted on 2005-02-21)
Some integers cannot be written as a sum of distinct positive squares. Does there exist a largest such integer? If so, find it.

 Submitted by David Shin Rating: 3.2857 (7 votes) Solution: (Hide) The answer is yes. The largest such integer is 128. To see why, we first prove a lemma. Lemma. Suppose n is a positive integer, and every integer from n+1 to 4n+35 (inclusive) can be written as a sum of distinct positive squares. Then every integer larger than n can be written as a sum of distinct positive squares. Proof. Suppose not, and let k be the least counterexample. By hypothesis, k>=4n+36. Write k in the form k=4q+r, with 0

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 Subject Author Date Answer K Sengupta 2008-03-21 15:01:57 re: solution David Shin 2005-03-01 19:14:31 solution Tom 2005-03-01 04:03:39 re: researched (spoiler) Richard 2005-02-22 17:22:39 re: meow David Shin 2005-02-22 16:00:26 meow Rex 2005-02-22 14:43:43 researched (spoiler) Charlie 2005-02-22 14:04:09 question Sachin 2005-02-22 06:46:16

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