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Consecutive Contemplation (Posted on 2012-11-08) Difficulty: 3 of 5
Each of the n consecutive positive integers x+1, x+2, ...., x+n is expressible as the sum of squares of two distinct positive integers.

Determine the maximum value of n and prove that no higher value of n is possible.

No Solution Yet Submitted by K Sengupta    
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Solution re: Not proven yet but I have a guess | Comment 3 of 6 |
(In reply to Not proven yet but I have a guess by Jer)


.. due to Fermat:
A number N is expressible as a sum of 2 squares if and only if in the prime factorization of N, every prime of the form (4k+3) occurs an even number of times!

Examples: 245 = 5*7*7. The only prime of the form 4k+3 is 7, and it appears twice. So it should be possible to write 245 as a sum of 2 squares (in fact, try the squares of 14 and 7). But because 7 appears only once in 21=3*7, it is impossible to write 21 as the sum of two squares.

SO: Since       4K+3  in odd  power CANNOT BE  a sum of 2 squares , two consecutive odd numbers are the sum of two distinct positive integers.- and 3 is the limit


Edited on November 9, 2012, 2:31 am
  Posted by Ady TZIDON on 2012-11-08 18:02:12

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