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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".

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  • 1
    $\begingroup$ 36729292763 18364646381 353, i found another example with Pari. $\endgroup$ – Enzo Creti Nov 7 '18 at 18:57
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    $\begingroup$ Approving the conjecture $\endgroup$ – Peter Nov 7 '18 at 18:57
  • 1
    $\begingroup$ @Barry Cipra another example found with Pari 70381018307 35190509153 727 $\endgroup$ – Enzo Creti Nov 7 '18 at 19:20
  • $\begingroup$ beginning with 12211811221, the prime factor of $p+1$ has the first and the last digit equal to the first and last digit of $p$. $\endgroup$ – Enzo Creti Nov 7 '18 at 20:13
  • $\begingroup$ beginning with 383, is factor(p-1)/factor(p+1) always increasing? $\endgroup$ – Enzo Creti Nov 7 '18 at 21:15
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There are two counterexamples not far away. The table continues:

              p   factor(p - 1)  factor(p + 1)  factor(p - 1)/factor(p + 1) 
              7              3              2               1.5000  
            383            191              3              63.6666  
          38783          19391            101             191.9900
    12211811221          30703            151             203.3311
    18345254381      917262719            101         9081809.0990
    36729292763    18364646381            353        52024493.9972
    70381018307    35190509153            727        48405101.9986
  1852347432581    92617371629    11434243411               8.0999
  1874989894781    93749494739          10301         9101009.0999
115582393285511          18181          77477               0.2346
164257606752461          73637        1245421               0.0591

The ratios in the final column are truncated to $4$ decimal places.

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