All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Numbers
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    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
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       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

 


  Posted by Charlie on 2012-11-08 22:31:52
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (7)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information