SOLUTION: Prove that the number {{{ (2n+1)^3-(2n-1)^3 }}} can be written as the sum of three squares.

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Question 1019845: Prove that the number +%282n%2B1%29%5E3-%282n-1%29%5E3+ can be written as the sum of three squares.
Found 2 solutions by LinnW, ikleyn:
Answer by LinnW(1048) About Me  (Show Source):
You can put this solution on YOUR website!
%282n%2B1%29%5E3 expands to 8n%5E3+%2B+12n%5E2+%2B6n+%2B+1
%282n11%29%5E3 expands to 8n%5E3+-+12n%5E2+%2B6n+-+1
So +%282n%2B1%29%5E3-%282n-1%29%5E3+ =
8n%5E3+%2B+12n%5E2+%2B6n+%2B+1 - 8n%5E3+-+12n%5E2+%2B6n+-+1
= 24n%5E2+%2B+2+
Check the answer using www.wolframalpha.com

Answer by ikleyn(52831) About Me  (Show Source):
You can put this solution on YOUR website!
.
Prove that the number +%282n%2B1%29%5E3-%282n-1%29%5E3+ can be written as the sum of three squares.
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Let me continue from the other post.

To complete the proof, we only need to represent 24n^2 + 2 as the sum of three square.

It is easy:

24n%5E2+%2B+2 = %284n%5E2+%2B+4n+%2B+1%29 + %284n%5E2+-+4n+%2B+1%29 + 16n%5E2 = %282n%2B1%29%5E2 + %282n-1%29%5E2 + %284n%29%5E2.

The statement is proved.