Primality testing gcd(V(52825),5*F(52825/5)*(F(52825/5)-1)+1) 
[N-1, Brillhart-Lehmer-Selfridge]
Calling Brillhart-Lehmer-Selfridge with factored part 30.13%
gcd(V(52825),5*F(52825/5)*(F(52825/5)-1)+1) is PRP! 

helper:

<<

2
5
5
13
641
1049
1087
1447
2113
4481
5737
35201
141961
2563969
75440641
75518783
683478401
1435815041
5072305921
11862575248703
42311526823103
5326379933318401
31187396775005101
35323750923943681
499750037222722721
8132307034170126721
3891324187650256896001
878132240443974874201601
2211745416606187093767661
1271150843287904563935054721
2811666624525811646469915877
471991644996070811213117005172115579281
1931734634522754726108707718410439174358483201
316837008400107576629115441641182686718784605974069121
717183675566658213132761730294626635650067479275923957352582406983301023453681347576121
1294884906033467167459275761658328869257948204777663121266918228225397498956629185213102721
4803658436325482520297638915610297327172500447085995837685448718393398021102387961508954460708151467171902227193383138649187081613761988158080851197141142687882714108512807667903716826727900953898257678338906293184183179605112525027159635875754366834111281530688122982068596616838092252817393837440415681649405023438489552891309442925601951853249448297320649244311189113417746777090338754586530687736559414352447922013221225094130264219311765429963842465899423437961862094906911217635020275701152671009954306655852270877554286118548877950785037990297343654313453616722792209319752996500195729949295106481009045526160574181325480600881237205398590084938129926524688840773271110628749178344879721

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with prime factors of N-1 proven by PFGW and Primo.
 
Then we used Konyagin-Pomerance,
since we have more than 30% of N-1.