solve in N²

May 2009
596
31
ALGERIA
solve in N²:
m² = 1+2ⁿ
 

SlipEternal

MHF Helper
Nov 2010
3,728
1,571
$$m^2-1 = 2^n$$

$$(m+1)(m-1) = 2^n$$

This implies that both $m-1$ and $m+1$ are both powers of $2$. However, there are no distinct powers of 2 that are a distance of 2 apart, so there are no solutions over the natural numbers.
 
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topsquark

Forum Staff
Jan 2006
11,578
3,454
Wellsville, NY
$$m^2-1 = 2^n$$

$$(m+1)(m-1) = 2^n$$

This implies that both $m-1$ and $m+1$ are both powers of $2$. However, there are no distinct powers of 2 that are a distance of 2 apart, so there are no solutions over the natural numbers.
m = 3, n = 3.

-Dan
 
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