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Zhi-Hong Sun 《Journal of Number Theory》2005,113(1):10-52
Let be a prime, m∈Z and . In this paper we obtain a general criterion for m to be a quartic residue in terms of appropriate binary quadratic forms. Let d>1 be a squarefree integer such that , where is the Legendre symbol, and let εd be the fundamental unit of the quadratic field . Since 1942 many mathematicians tried to characterize those primes p so that εd is a quadratic or quartic residue . In this paper we will completely solve these open problems by determining the value of , where p is an odd prime, and . As an application we also obtain a general criterion for , where {un(a,b)} is the Lucas sequence defined by and . 相似文献
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Let . We prove that a subset of , where p is a prime number, with cardinality larger than such that its subset sums do not cover has an automorphic image which is rather concentrated; more precisely, there exists s prime to p such that
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