Determine if a number is the sum of two triangular numbers.

Question

Is it possible to figure out if a number $z$ is the addition of two triangular number without recursion or finding the values to $x$ and $y$?

$$\frac{x(x+1)}{2} + \frac{y(y+1)}{2} = z$$

An example of this would be following:

$$\frac{345(345+1)}{2} + \frac{234(234+1)}{2} = 87180$$

Is it possible to determine if $87180$ originated from the addition of two triangular numbers without recursively going back and plugging every possibility for $x$ and $y$?

Answer 1

The equation: $$ x(x+1)+y(y+1)=2z \tag{1}$$ is equivalent to: $$ (2x+1)^2 + (2y+1)^2 = 8z+2 \tag{2} $$ hence $z$ is the sum of two triangular numbers iff $8z+2$ is the sum of two squares, i.e. iff for every prime $p$ of the form $4k+3$ that divides $8z+2$, $\nu_p(8z+2)$ is even.

In the given example, $z=87180$, we have:

$$ 8z+2 = 2\cdot 17 \cdot 73\cdot 281 \tag{3}$$ and every odd prime occurring in the RHS of $(3)$ is of the form $4k+1$, hence $8z+2$ can be written as the sum of two odd squares: $$ 8z+2 = 101^2 + 829^2 \tag{4}$$ and $z$ is the sum of two triangular numbers: $$ 87180 = \binom{51}{2}+\binom{415}{2}.\tag{5} $$

Answer 2

If we had $a(k)$ from A008441 the test for $z$ would be $$ a(z) > 0 . $$

The information section on this series provides many interesting relations.