What is the smallest prime number n which can be formulated with all of the three following integer equations? $$u^2 + 4^2 \cdot v^2 = n$$ $$w^2 + 9^2 \cdot x^2 = n$$ $$y^2 + 11^2 \cdot z^2 = n$$
closed as off-topic by Xander Henderson, Saad, José Carlos Santos, clathratus, Paul Frost Jan 1 at 21:54
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Using the Theorem:
If a prime can be expressed as sum of two squares then the representation is unique. Proof
Then we have that the three expressions are the same.
Then for minimize $n$ we need find a prime that can be expressed as any of this forms: $$4^2\cdot 11^2 r^2+9^2 j^2$$ or $$4^2\cdot 9^2 l^2+11^2 f^2$$ But $36^2+11^2=1417$ isn't a prime and by coincidences of the life $44^2+9^2=2017$ is a prime number. We are done.
The solution is easy to write.
Here the representation of 3 options, but it is easy to see that can be written in the form of a combination with any number of options.