On primes in Lucas sequences

Denote the Fibonacci and Lucas numbers by.; F0=2,F1=1,and Fn=Fn-1+Fn- 2, forn≥2 and L0=2,L1=1, andLn=Ln-1 +Ln-2, forn≥2. We will study conditions under which a prime p ≡ 1 (mod 4) divides Lp-1/4 or Fp-1/4 . In particular, we will prove the following theorem.;Theorem. Let p be an odd prime and i=-1 . (a) If p ≡ 1 (mod 40) or p ≡ 9 (mod 40), then p∣Lp-1/4 if and only if 1+2ip-1 /22-i +1-2ip- 1/22+i ≡-4 (mod p) and p∣Fp-1/4 if and only if 1+2ip-1 /22-i +1-2ip- 1/22+i ≢-4 (mod p). (b) If p ≡ 21 (mod 40) or p ≡ 29 (mod 40), then p∣Lp-1/4 if and only if 1+2ip-1 /22-i +1-2ip- 1/22+i ≡4 (mod p) and p∣Fp-1/4 if and only if 1+2ip-1 /22-i +1-2ip- 1/22+i ≢4 (mod p).;We will conclude this thesis with some examples and some open questions.