Split It As Two Squares?

Number Theory Level 2

If n n is a positive integer such that 2 n + 1 2n+1 is a perfect square , is it true that n + 1 n+1 is the sum of two consecutive perfect squares?

Yes No

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Anne Beatriz by Anne Beatriz , 18, Brazil

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2 solutions

Because 2 n + 1 2n+1 is odd, there is a positive integer k k such that:

2 n + 1 = ( 2 k + 1 ) 2 \quad2n+1=(2k+1)^2

or 2 n + 1 = 4 k 2 + 4 k + 1 2n+1=4k^2+4k+1

or n = 2 k 2 + 2 k n=2k^2+2k

or n + 1 = k 2 + ( k + 1 ) 2 n+1=k^2+(k+1)^2

So, the answer must be YES .

36 Helpful 0 Interesting 1 Brilliant 0 Confused

Nice solution.. Upvoted.. : ) \ :)

Parag Zode - 5 years, 2 months ago
Finn Hulse
Mar 23, 2016

Working from the bottom up, we see that if there is some k k such that it squared plus k + 1 k+1 squared is half of 1 more than a perfect square. Thus we get the equation

2 n + 1 = 4 k 2 + 4 k + 1 2n+1=4k^2+4k+1

And we can see that 4 k 2 + 4 k + 1 4k^2+4k+1 is the square of 2 k + 1 2k+1 , thus the answer is yes.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Can me explain what dya mean by half of '1 more than a perfect square' or could you write that out in equation+

Zoro Marak - 5 years, 2 months ago

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