a) If $p\ne 5$, then $p^2\equiv \pm1\pmod 5$, so $4p^2+1,6p^2+1\equiv 0,2\pmod 5$

part (b) is trivial; simply add 1 to each equation, multiply them all, take cube roots, and divide this by each equation individually.

Part (b): Since the last three equations are highly symmetric, it is natural to look for a solution with $x=y=z$. The first equation shows that $x=y=z=1$ might work. With that, the other equations all turn into $4u+3=10$, so that $u=7/4$.

TheDarkPrince wrote: (a) Find all prime numbers $p$ such that $4p^2+1$ and $6p^2+1$ are also primes. (b)Find real numbers $x,y,z,u$ such that \[xyz+xy+yz+zx+x+y+z=7\]\[yzu+yz+zu+uy+y+z+u=10\]\[zux+zu+ux+xz+z+u+x=10\]\[uxy+ux+xy+yu+u+x+y=10\] How was todays NMTC?

$For part a we get that pis not congruent to 1 or -1 mod 4 for 4p^2 +1 not to be divisible by 5. And p is not congruent to 2 or 3 in second condition which gives 5/p Only such prime is p=5.$

Part B Simplifying equations you get, either u=-1 or x=y=z.after that simple algebra.....

what is the correct proof to part a?

The solution to part a . Look at 4p^2 +1 modulo 5 ,u will get that it is either congruent to 0 or 2 .Not examining p=5, we conclude that it should be congruent to 2 mod 5 since it is a prime.But this case arises when itself is congruent to 2,3 mod 5. But using these values in 6p^2 +1,we get it to be congruent to 0 mod 5,which is certainly not a prime.Hence no solutions exist when p is not equal to 5,Examining p equal to 5 ,we get it to be our only solution.

The solution to part a . Look at 4p^2 +1 modulo 5 ,u will get that it is either congruent to 0 or 2,when p itself is not equal to 5. We conclude that it should be congruent to 2 mod 5 since it is a prime.But this case arises when p itself is congruent to 2 or 3 mod 5. But using these values in 6p^2 +1,we get it to be congruent to 0 mod 5,which is certainly not a prime.Hence no solutions exist when p is not equal to 5.Examining p equal to 5 ,we get it to be our only solution. Sorry I find it difficult to use latex hence couldn"t show the working.

thebigdog wrote: $For part a we get that pis not congruent to 1 or -1 mod 4 for 4p^2 +1 not to be divisible by 5. And p is not congruent to 2 or 3 in second condition which gives 5/p Only such prime is p=5.$ For part a we get that pis not congruent to $1$ or $-1 \pmod 4$ for $4p^2 +1$ not to be divisible by $5$. And p is not congruent to 2 or 3 in second condition which gives $\frac{5}{p}$ Only such prime is p=5.