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 Single or double (Posted on 2010-12-24)
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.

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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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