A prime number is called a Wilson prime
if p^2 divides (p − 1)! + 1,
There are only 3 known Wilson primes:
5, 13 and ...

Find the 3rd one.

Hint : It is a 3-digit number.

   10   P=1
   20   while P<1000
   30    P=nxtprm(P)
   40    Dvsr=P*P
   50    Dvnd=!(P-1)+1
   60    if Dvnd @ Dvsr=0 then print P,Dvsr,Dvnd
   70   wend

finds

 

 p       p^2     (p-1)! + 1
 5       25      25
 13      169     479001601
 563     316969          1128062102650147335604931877600826356732599785899156109
94879922067851216850776523498248316738375390580914470613504049122882247526761662
32504987238597734405218500241472929420765736368417076000786819232752956209532030
17526825519806548991507659470422441620802456355592966694050152826017764480535088
45595778363500203282317374589290155381842194432599017760692680028609741462705405
18645360418702282857215146853858602923779882799017519412252775788781353568993148
42081375921809281861368342897974853503506053425548001995956430815735416216069383
39158786368256259399736527412273663575936188904743783303502232789030651288843466
60250635861534373558801254128426232537595947202821030418565781987924866956019256
41400233914291862551103903663877005347420624372478858572341037671247912700942422
55868733069879313831280372547776338347570810050764225920596047369220167881984895
89238019374441997355890966064057477037099668260531617003866321836270934196409597
47268432757155219792503683484357034710631326735554924505495729303122508125116263
70105274275251208167297520077950422716994917370423246532157939503284320016474981
17946267801978081671446223063712925300632803498973864795497173769781248000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000001

so 563 is the answer.

Posted by Charlie on 2012-08-07 16:01:19