Are there any well known patterns about which finite simple groups have order $ p+1 $ for $ p $ a prime?
Here is a list of all non-cyclic simple groups of order up to 100,000 and whether they have order p+1 (there are 31 such groups, 16 have order $ p+1 $)
$ PSL_2(5) $, $p=59$
$ PSL_2(7) $, $p=167$
$ PSL_2(9) $, $p=359$
$ PSL_2(8) $, $p=503$
$ PSL_2(11) $, $p=659$
$ PSL_2(13) $, $p=1091$
$ PSL_2(17) $, $p=2447$
$ A_7 $, $2519$ not prime
$ PSL_2(19) $, $3419$ not prime
$ PSL_2(16) $, $p=4079$
$ PSL_3(3) $, $5615$ not prime
$ PSU_3(3) $, $p=6047$
$ PSL_2(23) $, $6071$ not prime
$ PSL_2(25) $, $7799$ not prime
$ M_{11} $, $p=7919$
$ PSL_2(27) $, $9827$ not prime
$ PSL_2(29) $, $12,179$ not prime
$ PSL_2(31) $, $p=14,879$
$ PSL_4(2) $, $20,159$ not prime
$ PSL_3(4) $, $20,159$ not prime
$ PSL_2(37) $, $p=25,307$
$ PSU_4(2) $, $p=25,919$
$ Suz(8) $, $29,119$ not prime
$ PSL_2(32) $, $32,735$ not prime
$ PSL_2(41) $, $p=34,439$
$ PSL_2(43) $, $p=39,731$
$ PSL_2(47) $, $51,887$ not prime
$ PSL_2(49) $, $58,799$ not prime
$ PSU_3(4) $, $62,399$ not prime
$ PSL_2(53) $, $p=74,411$
$ M_{12} $, $95,039$ not prime