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.
(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 2
13 2 3
17 1 4
25 3 4
29 2 5
37 1 6
41 4 5
45 3 6
53 2 7
61 5 6
65 1 8
73 3 8
85 2 9
89 5 8
97 4 9
101 1 10
109 3 10
113 7 8
117 6 9
125 2 11
137 4 11
145 1 12
149 7 10
153 3 12
157 6 11
169 5 12
173 2 13
181 9 10
185 4 13
193 7 12
197 1 14
205 3 14
221 5 14
225 9 12
229 2 15
233 8 13
241 4 15
245 7 14
257 1 16
261 6 15
265 3 16
269 10 13
277 9 14
281 5 16
289 8 15
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Posted by Charlie
on 2012-11-08 22:31:52 |
re(2): Not proven yet but I have a guess
