Dear Sir, No solution in integer not all zero. Regards, Peeyush

well , usually you should find a solution , anyways , let $a=c$ so $(a-b)^2=2a+b$ letting $a=b+t$ we have $t^2=3b+2t$ so $b=\frac{t^2-2t}{3}$ these work if they are integers now if $ a>b>c$ and $a=x+y+c,b=x+c$ we have $\frac{x^2+y^2+(x+y)^2}{2}=3c+2x+y$ so $x^2+y^2+xy=3c+2x+y$ and again $c=\frac{x^2+y^2+xy-2x-y}{3} $ works , so these work if they are integer aswell.

I'm not sure if there is a mistake. As the equation is symmetric in $a,b,c$. WLOG $a \le b,c$ Therefore, there are non-negative $m,n$ such that $b = a+m, c = a+n$ By substitution and simplification, the equation becomes $m^2+n^2=mn+m+n+3a$. So $a=\frac{m^2+n^2-mn-m-n}{3}=\frac{(m+n)^2 - (m+n) - 3mn}{3}$ By case work with modulo 3 on $m,n$, there are 6 cases out of 9 cases such that $a$ is an integer: 1. $m \equiv 0 \pmod{3},n \equiv 0 \pmod{3}$ 2. $m \equiv 0 \pmod{3},n \equiv 1 \pmod{3}$ 3. $m \equiv 1 \pmod{3},n \equiv 0 \pmod{3}$ 4. $m \equiv 1 \pmod{3},n \equiv 2 \pmod{3}$ 5. $m \equiv 2 \pmod{3},n \equiv 1 \pmod{3}$ 6. $m \equiv 2 \pmod{3},n \equiv 2 \pmod{3}$ (which is equivalent to $m+n \not\equiv 2 \pmod{3}$ derived from the second expression). To check whether $a$ is non-negative, $\frac{1}{2}[(m-n)^2+(m-1)^2+(n-1)^2] \ge 0$ $m^2+n^2-mn-m-n \ge -1$ , equality holds if and only if $m=n=1$ which isn't possible as $m+n \not\equiv 2 \pmod{3}$ Hence, $m^2+n^2-mn-m-n > -1$, LHS is an integer so $m^2+n^2-mn-m-n \ge 0$ and $a \ge 0$ as desired. All solutions are $(a,b,c)=(\frac{m^2+n^2-mn-m-n}{3},\frac{m^2+n^2-mn+2m-n}{3},\frac{m^2+n^2-mn-m+2n}{3})$ with non-negative integers $m,n$ such that $m+n \not\equiv 2\pmod{3}$ and their permutations

PeeyushPandaya wrote: Dear Sir, No solution in integer not all zero. Regards, Peeyush You're not the first person to make such a big mistake. https://artofproblemsolving.com/community/c3046h1915865_combination_of_three https://math.stackexchange.com/questions/3350399/solve-diophantine-equation-with-three-variables-part-two

$$a,a+x,a+x+y$$$$3a=x^2+y^2+xy-2x-y$$