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Single or double (Posted on 2010-12-24) Difficulty: 3 of 5
Both a number N and its half,when augmented by 1 (i.e.N+1 and N/2+1) form a perfect square.
0 is a trivial example.

List 4 more examples.

No Solution Yet Submitted by Ady TZIDON    
Rating: 4.0000 (1 votes)

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Similar approach | Comment 3 of 6 |

1 Call n+1, x^2; call 1/2n+1, y^2     
2 Then x^2*y^2=(n+1)(n/2+1)     
 (Incidentally, numbers exactly one less than A001108 in Sloane)     
3 Looking at RHS     
 (n+1)(n/2+1) = n^2/2+3n/2+1     
 So, 2(xy)^2=n^2+3n+2     
 A Pellian, with positive integer solutions at      
 n = 1/4 ((3-2*(2)^(1/2))^m+(3+2*(2)^(1/2))^m-6)       
 Of which exactly half (those with odd m)     
 are also solutions to the initial problem     
4 Write     
 n = 1/4 ((3-2*2^(1/2))^(2m-1)+(3+2*2^(1/2))^(2m-1)-6)       
5 Then:     
 m n x y  
 2 48 7 5  
 3 1680 41 29 
 4 57120 239 169 
 5 1940448 1393 985 
 6 65918160 8119 5741 
 7 2239277040 47321 33461 
 8 76069501248 275807 195025 
 9 2584123765440 1607521 1136689 
 10  87784138523760 9369319 6625109 
 provides the first few positive solutions

Compare Sloane A008845 A002315 A001653  


  Posted by broll on 2010-12-25 01:02:58
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