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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

Edit: cross-posted to MO here https://mathoverflow.net/questions/431254/finite-simple-groups-of-order-p1 see also https://mathoverflow.net/questions/48618/how-many-finite-simple-groups-of-order-p1?noredirect=1&lq=1

Ian Gershon Teixeira
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  • First thought: there are probably more such (nonabelian) simple groups than the PNT would predict, thanks to heuristics about how many divisors $p+1$ has (https://math.stackexchange.com/questions/3491929/why-do-even-numbers-which-surround-primes-have-more-divisors-than-those-which-su). – Ravi Fernando Sep 07 '22 at 22:08
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    Since most families of finite simple groups have polynomial order in the size of an underlying field, you could probably prove that there exist infinitely many examples (and estimate their frequency) conditionally on the Bateman-Horn conjecture. The case of $A_n$ is probably harder to say things about. But I suspect that the only unconditional answer you'll get is for the groups of prime order. – Ravi Fernando Sep 07 '22 at 22:16
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    The alternating groups $A_n$ for $n=5,6,9,31,41,373,589$ are of this form, and these are the only ones up to $n=800$. – David A. Craven Sep 08 '22 at 21:42
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    And $n=812,989$ up to $1000$. Looks like there might be infinitely many such $n$. The latter of these has 2534 digits and ends with 240 9s. – David A. Craven Sep 08 '22 at 22:12
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    @DavidA.Craven http://oeis.org/A082671 – Brauer Suzuki Sep 09 '22 at 05:04
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    @BrauerSuzuki It never ceases to amaze me what sort of things people have uploaded to that database. – David A. Craven Sep 09 '22 at 10:14
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    Of the simple non-abelian groups up to order $10^{20}$, there are $40007$ orders of the form $p+1$, for prime $p$. The only pattern I've seen is that the vast majority are $\operatorname{PSL}(2,q)$s, which is what one would expect. The largest non-PSL in that list is the Janko group $J_4$ of order $86775571046077562879 + 1$. – James Sep 09 '22 at 20:50
  • @James how many simple non-abelian groups of order up to $ 10^{20} $ are there? (just to get a sense of $ 40,007 $ orders of the form $ p+1 $ out of how many) – Ian Gershon Teixeira Sep 09 '22 at 20:59
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    @IanGershonTeixeira The number of orders of simple non-abelian groups up to $10^{20}$ is $403864$, but some of those are the order of two distinct groups. If my script is correct, that would add another $6$ actual groups to that number. – James Sep 09 '22 at 21:18
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    Are you interested in some more primes of the form $\frac{n!}{2}-1$ ? Slight correction , for $n=989$ , we get a $2535$-digit prime ending with $244$ nines. here are the next ones. – Peter Oct 28 '22 at 13:52
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    this one is already really large. – Peter Oct 28 '22 at 13:58
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    and this one seems to be the largest one factordb knows. – Peter Oct 28 '22 at 14:02

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