$a$ is an odd number and the value of $a$ could be between $1$ and $39$.
Find the value of $a$ which means the equation:
$$x^2 +2xyz +y^2 =a$$
has integer roots
Thanks in advance!
2026-04-24 04:02:18.1777003338
Find the value of a which form the equation
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In other words find all values $a$ which are attainable by function $f=x^2+2xyz+y^2$ if $x,y,z$ are integers. Take $x=y=1$ then $f=2z+2$ and so all even numbers are attainable. Take $x=1, y=2$ then $f=4z+5$ and so all the integers congruent to $1\pmod{4}$ are attainable. Show that the rest integers are not attainable. As they are odd then only one of $x$ and $y$ must be odd. But in this case we would have $f\equiv 1\pmod {4}.$ In a given range only 10 numbers $3,7,\ldots, 39$ are not attainable.