# New Year 2017 Puzzle: Two of Four Diophantine Equations [closed]

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, ShaileshDec 27 '16 at 1:45

This question appears to be off-topic. The users who voted to close gave these specific reasons:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – John B, Shailesh
• "This question is not about mathematics, within the scope defined in the help center." – Will Jagy, Namaste
If this question can be reworded to fit the rules in the help center, please edit the question.

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

One solution: $$a=b=7, \quad c=11$$ $$d=13, \quad e=43$$ and, quite fittingly, $$n=2017.$$
• 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. – Carl Schildkraut Dec 27 '16 at 0:39