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?
(In reply to
re: Solution by Steve Herman)
Good
to get a response Steve.
In my defence, I didn’t say I was splitting the LHS into a
product of odd and even numbers; yes that might be possible
in many ways. I was splitting it into two squares – that
can only be done in one way, since the consecutive numbers
a and a + 1, and therefore their squares, can share no prime
factors. So the fact that these squares appear in the equations
denies the possibility of any ‘other ways’.
For example, since 2 cannot be a factor of consecutive integers,
the two parts a^2 and (a + 1)^2 cannot both be even.
Does this fill the holes in, or will I have to climb into one.

Posted by Harry
on 20150112 11:09:08 