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Numbers of the form $x^2(x-1) + y^2(y-1) + z^2(z-1)$ with $x,y,zinmathbb Z$

Roger Heath-Brown conjectured that any integer $knotequivpm4pmod9$ can be expressed as $x^3+y^3+z^3$ with $x,y,zinmathbb Z$ in infinitely many different ways. Also it is well-known that some solutions are extremely difficult to construct and currently there are interesting and notable discoveries in that direction such as representations of 33 and 42.

I am not sure that my question is relatively easy. It is related to https://oeis.org/A336205. I conjecture that any even integer can be represented as $x^2(x-1) + y^2(y-1) + z^2(z-1)$ with $x, y, zinmathbb Z$. I also conjecture that this is possible in infinitely many different ways for any even integer. In particular, there can be some related results on that direction since I easily formulated below parametric one based on comment of Robert Israel at https://oeis.org/A336240. $$x^2(x-1) + y^2(y-1) + z^2(z-1) = 2a^2(a-1)$$ with $x = 1 – (6a-2)m^2$, $y = a – m(1-(6a-2)m^2)$, $z = a + m(1-(6a-2)m^2)$. I don’t know that there are some references on that equation or further similar results. If someone can see additional nontrivial ones, comments on parametric solutions are also very welcome.

It is obvious that first conjecture can be rewritten based on definition of generalized pentagonal pyramidal numbers, i.e., any integer can be expressed as $xT_x + yT_y + zT_z$ with $x,y,zinmathbb Z$ where $T_n$ is nth triangular number.

It is relatively easy to find solutions to this spesific representation, at least for any even integer $t$ with $-1000 leq t leq 1000$. Is it well-known fact or conjecture ? If this is not known, can someone prove this conjecture ? (Also more experimental evidences would be very welcome if proof is probably very hard.)

Finally, I would like to share this note kindly. Although I am not sure that it is sufficiently interesting or meaningful, I also believe that it is probable to construct further questions and related generalized conjectures to some sort of similar representations such as $f(x)T_x + f(y)T_y + f(z)T_z$ or $g(x)x^2 + g(y)y^2 + g(z)z^2$ based on several natural selections of $f$ and $g$. For example, $g(w)=w-2$ could be fruitful or investigation of sum of three generalized $r$-gonal pyramidal numbers can provide interesting experimental results for certain values of $r$ : https://math.stackexchange.com/questions/2472205/is-every-number-a-sum-of-3-tetrahedral-numbers and Does the set ${binom x3+binom y3+binom z3: x,y,zinmathbb Z}$ contain all integers? ($r = 3$).

MathOverflow Asked on November 22, 2021

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