<s>[INST] A question about the number 541456 I found that the concatenation in base 10 of $2^{541456}-1$ and $2^{541455}-1$, gives a probable prime.
I also found about number $541456$ that:
$$5\cdot(5^2+4^2+1+4^2+5^2+6^2)=(5^3+4^3+1+4^3+5^3+6^3)$$
the equation $a\cdot(2\cdot a^2+2\cdot b^2+c^2+d^2)=(2\cdot a^3+2\cdot b^3+c^3+d^3)$ besides $a=5, b=4, c=6, d=1$ has other non trivial solutions with a,b,c,d positive integers?
 [/INST] Your equation $a*(2*a^2+2*b^2+c^2+d)=(2*a^3+2*b^3+c^3+d)$ is equivalent to
$$2ab^2+ac^2+ad=2b^3+c^3+d$$
I find the below solutions assuming all of the variables are in the range $1$ to $9$.  I just did a search.

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