Are all these pairs of primes twin primes? For $a$ and $b$ primes, if both $(a^b \bmod b)$ and $(b^a \bmod a)$ are prime,
does this imply that $(a,b)$ are twin primes?
For example, for $(a,b)=(41,43)$, $(41^{43} \bmod 43) = 41$ and $(43^{41} \bmod 41) = 2$.
Whereas for $(a,b)=(41,47)$, $(41^{47} \bmod 47) = 41$
but $(47^{41} \bmod 41) = 6$, not a prime.

No. $3^{11} \bmod 11 = 3$, and $11^3 \bmod 3 = 2$, but $3$ and $11$ are not twin primes.
$(5,13)$ also works, and both of the other primes constructed in this case are odd.
