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 T times T creates another T?? (Posted on 2015-01-11)
The squares of triangular numbers 1 and 6 are triangular numbers 1 and 36.

T1^2 = 1 * 1 = 1 = T1
T3^2 = 6 * 6 = 36 = T8

Are there additional triangular numbers whose squares form a triangular number?

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 Solution | Comment 1 of 8
[a(a + 1)/2]2 = b(b + 1)/2  simplifies to   a2(a + 1)2 = 2b(b+1).

The LHS is a product of two squares, one even the other odd.
The RHS can be split into odd and even parts in two different
ways: 2b*(b + 1)  or  b*2(b + 1), so that four pairs of
equations are possible:

(1)          a2 =  2b  &  (a + 1)2 =  b + 1             giving  a = 0 or -4,

(2)          a2 = b     &  (a + 1)2 = 2(b + 1)          giving  a = 1,

(3)          a2 = b + 1  &  (a + 1)2 = 2b               giving  a = 3 or -1,

(4)          a2 = 2(b + 1)  &  (a + 1)2 = b            giving  a = -2.

a and b must be positive integers, so the only two
possibilities are those given in the question:

a = 1,  b = 1:    T12 = T1      and     a = 3,  b = 8:    T32 = T8

 Posted by Harry on 2015-01-11 11:53:37

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