What is the smallest number $n$ which can be described with equation $(1)$ and one or more of equations $(2.1)-(2.3)$, where all variables are prime numbers? $$a^3 + b^3 + c^3 = n \tag{1}$$ $$d^2 + e^2 -1 = n \quad\tag{2.1}$$ $$f^2 + f \cdot g + g^2 = n \tag{2.2}$$ $$h = n^4 + (n-1)^4 + (n+1)^4\tag{2.3}$$

Is there a second solution for all four equations together? I let the computer check the first million of primes without success.


closed as off-topic by Will Jagy, C. Falcon, Namaste, John B, Shailesh Dec 27 '16 at 1:45

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  • $\begingroup$ When you say "prime number equations", do you mean to say that $a,b,c$ are primes? $\endgroup$ – Arthur Dec 26 '16 at 23:55
  • $\begingroup$ Does $n$ have to be a prime as well? $\endgroup$ – Carl Schildkraut Dec 27 '16 at 0:32

One solution: $$a=b=7, \quad c=11$$ $$d=13, \quad e=43$$ and, quite fittingly, $$ n=2017. $$

  • 1
    $\begingroup$ This solution also has $n^4+(n-1)^4+(n+1)^4=49652974982033$ which is prime, and $n=41^2+41\cdot 7 + 7^2$. Thus, it satisfies all of the equations. $\endgroup$ – Carl Schildkraut Dec 27 '16 at 0:39

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