[seqfan] Re: Two elusive triangular numbers

[Matthijs Coster]

Hello Ant,

Your question can be replaced by another question about the existence of 
a set of elliptic curves.
First step is to replace the problem by:
Find intebers u,v,w and t such that u(u+1) + t(t+1) = v(v+1) and v(v+1) 
+ t(t+1) = w(w+1).
Multiply these equations by 4 and add 2 at both sides we get:
(2u+1)^2 + (2t+1)^2 = (2v+1)^2 + 1 and (2v+1)^2 + (2t+1)^2 = (2w+1)^2 + 1.
Now replace 2u+1 by a; 2v+1 by b; 2w+1 by c and 2t+1 by z. We get:
a^2 + z^2 = b^2 + 1 and b^2 + z^2 = c^2 + 1. We see easily that a^2 + 
c^2 = 2b^2.
Therefore b has to be written as b = x^2 + y^2. We may assume x > y. It 
follows easily that a and c can be expressed in terms of x and y. We get:
a = x^2 - 2xy - y^2 and c = x^2 + 2xy - y^2 and z^2 = c^2 - b^2 + 1 = 
b^2 - a^2 + 1 = 4xy(x^2 - y^2) + 1.

Therefore the original question follows from finding all integral 
solutions of z^2 = 4xy(x^2 - y^2) + 1.

There is an infinitely list of (trivial) solutions based on 
http://oeis.org/A001653 (Numbers n such that 2*n^2 - 1 is a square): 5, 
29, 169, 985, 5741, 33461, 195025, 1136689, ...  .  This sequence gives 
b's such that 2b^2 = odd square + 1.
For these numbers there is a set of equations in triangular numbers T(u) 
+ T(t) = T(v) and T(v) + T(t) = T(w):
5: 0 + 3 = 3 and 3 + 3 = 6 (T(0) + T(2) = T(2) and T(2) + T(2) = T(3))
29: 0 + 105 = 105 and 105 + 105 = 210 (T(0) + T(14) = T(14) and T(14) + 
T(14) = T(20))
169: 0 + 3570 = 3570 and 3570 + 3570 = 7140 (T(0) + T(84) = T(84) and 
T(84) + T(84) = T(119))
etc.

For other solutions except of {T(3) + T(5) = T(6) and T(6) + T(5) = 
T(8)} and {T(11) + T(14) = T(18) and T(18) + T(14) = T(23)} I think you 
have to consider z^2 = 4xy(x^2 - y^2) + 1. For fixed y this is the 
equation of an elliptic curve, and therefore has a finitely number of 
solutions (maybe 0). But it is unclear how many solutions there are in 
total.

--Matthijs Coster

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At 22-12-2010 19:22, mathstutoring wrote:
> The triangular numbers T(14)=105 and T(18)=171 have the property that their sum, difference and product are all triangular numbers.
>
> Does anyone know of the existence of a pair of triangular numbers T(m) and T(n) such that their sum, difference, product and quotient are also triangular numbers.
>
> Or alternatively know of the existence of a proof that no such m and n exist.
>
> Ant
>
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