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- W2034308822 abstract "Let p ≡ 1 ( mod 4 ) be a prime and a , b ∈ Z with a 2 + b 2 ≠ p . Suppose p = x 2 + ( a 2 + b 2 ) y 2 for some integers x and y . In the paper we develop the calculation technique of quartic Jacobi symbols and use it to determine ( b + a 2 + b 2 2 ) p − 1 4 ( mod p ) . As applications we obtain the congruences for U p − 1 4 modulo p and the criteria for p | U p − 1 8 (if p ≡ 1 ( mod 8 ) ), where { U n } is the Lucas sequence given by U 0 = 0 , U 1 = 1 and U n + 1 = b U n + k 2 U n − 1 ( n ⩾ 1 ) . We also pose many conjectures concerning U p − 1 4 , m p − 1 8 or m p − 5 8 ( mod p ) ." @default.
- W2034308822 created "2016-06-24" @default.
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- W2034308822 date "2009-03-01" @default.
- W2034308822 modified "2023-09-28" @default.
- W2034308822 title "Quartic, octic residues and Lucas sequences" @default.
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- W2034308822 doi "https://doi.org/10.1016/j.jnt.2008.09.019" @default.
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