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

Home > Numbers
Any others? (Posted on 2013-06-18) Difficulty: 2 of 5
100 is a square which is also a sum of 4 consecutive cubes.
Are there any others?

No Solution Yet Submitted by Ady TZIDON    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution | Comment 1 of 9
Let's call the cubes (x),(x+1),(x+2),(x+3)
The simplified sum is 4x+18x+42x+36
this factors to 2(2x+3)(x+3x+6)
this can only be a square if
2(2x+3)=(x+3x+6) or (2x+3)=2(x+3x+6)
The first of these is the quadratic 0=x-x which has solutions x=0 and x=1
The second is the quadratic 2x+4x+9=0 which has complex solutions.

So there are two squares that are the sum of 4 consecutive cubes:

0+1+2+3=36=6
1+2+3+4=100=10


  Posted by Jer on 2013-06-18 12:11:52
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 (1)
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