A curious case of $1729$

If I understood correctly the problem is

Find numbers $n$ such that there exist a pair $(a,b)$ with the property that

$a^3+b^3=n$

$a^2+b^2$=products of the digits of n + Sum of the digits of n

Then, the number $1547$ is a solution, for the pair $(a,b)=(11,6)$:

$11^3+6^3=1547\\11^2+6^2=1\cdot 5 \cdot 4 \cdot 7 + 1 + 5 + 4 + 7\\11^1+6^1 = 1+5+4+7$

And of course also the number $0$ is a solution. I also feel that there are no more solutions.


The first two conditions are satisfied by $$a=6\ \ \ \ , \ \ \ \ b=11$$


$(a,b)=(12,1)$ and $(a,b)=(11,6)$ are the only integer solutions with $2500 \geq a > b$.

Tags:

Puzzle

Number Theory

Recreational Mathematics

Problem Solving

Elementary Number Theory

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