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 Consecutive Contemplation (Posted on 2012-11-08)
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 No Rating

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 re(2): Not proven yet but I have a guess | Comment 4 of 6 |
(In reply to re: Not proven yet but I have a guess by Ady TZIDON)

You say

"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! "

But 49 is factored as 7*7 so that the 7 occurs an even number of times but 49 is not expressible as the sum of 2 squares.

Also:

"SO: 4K+1 CANNOT BE  a sum of 2 squares , THEREFORE..no two consecutive odd numbers are the sum of two distinct positive integers."

In the below list the number on the left is of the form 4k+1 and is expressible as the sum of the squares of the numbers to their right, such as 13=2^2 + 3^2:

`5       1       213      2       317      1       425      3       429      2       537      1       641      4       545      3       653      2       761      5       665      1       873      3       885      2       989      5       897      4       9101     1       10109     3       10113     7       8117     6       9125     2       11137     4       11145     1       12149     7       10153     3       12157     6       11169     5       12173     2       13181     9       10185     4       13193     7       12197     1       14205     3       14221     5       14225     9       12229     2       15233     8       13241     4       15245     7       14257     1       16261     6       15265     3       16269     10      13277     9       14281     5       16289     8       15`

 Posted by Charlie on 2012-11-08 22:31:52

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