Suppose, $p\ge 7$ is a palindrome-prime , the largest prime factor of $p-1$ is a palindrome-prime and the largest prime factor of $p+1$ is also a palindrome-prime.
Must the prime factor of $p-1$ be larger than the prime factor of $p+1$ ?
The first few solutions are :
p factor(p - 1) factor(p + 1)
7 3 2
11 5 3
383 191 3
38783 19391 101
12211811221 30703 151
18345254381 917262719 101
Beginning with $383$, the prime factor of $p-1$ is even vastly larger than that of $p+1$, but this could be a case of the "law of small numbers".
Best Answer
There are two counterexamples not far away. The table continues:
The ratios in the final column are truncated to $4$ decimal places.