Found with PFGW Version 3.7.5

373^5162-5162^373 is 2-PRP! (5.1663s+0.0020s)
373^5162-5162^373 is 3-PRP! (4.8469s+0.0021s)
373^5162-5162^373 is 5-PRP! (4.7372s+0.0016s)
373^5162-5162^373 is 7-PRP! (4.6039s+0.0016s)
373^5162-5162^373 is 11-PRP! (5.2278s+0.0023s)

Primality testing 373^5162-5162^373 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N+1 test using discriminant 11, base 1+sqrt(11)
Calling N-1 BLS with factored part 0.08% and helper 0.03% (0.27% proof)
373^5162-5162^373 is Fermat and Lucas PRP! (26.0701s+0.0033s)

To be completed...