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Quartic residues and binary quadratic forms. (English) Zbl 1101.11003

Let \(p\equiv 1\pmod 4\) be a prime and \(m\) an integer such that \(p\) does not divide \(m\). The basic problem of quartic residues is to characterize those primes \(p\) for which \(m\) is a quartic residue modulo \(p\). A very celebrated result in this direction being the result of Gauss that 2 is a quartic residue modulo \(p\) if and only \(p=x^2+64y^2\) for some integers \(x\) and \(y\). The author solves the problem for general \(m\) by giving a similar result, but with \(p\) being represented by one from a set of binary quadratic forms.
Let \(d>1\) be a squarefree integer and \(\epsilon_d\) be the fundamental unit of the quadratic field \(\mathbb Q(\sqrt{d})\). Suppose that \(p\equiv 1\pmod 4\) is a prime such that \((d/p)=1\) (the Legendre symbol). Many mathematicians tried to characterize primes of this form such that \(\varepsilon_d\) is a quadratic residue modulo \(p\). For example, A. Aigner and H. Reichardt [J. Reine Angew. Math. 184, 158–160 (1942; Zbl 0027.01102)], and independently P. Barrucand and H. Cohn [J. Reine Angew. Math. 238, 67–70 (1969; Zbl 0207.36202)] proved that \(\varepsilon_2=1+\sqrt{2}\) is a quadratic residue of a prime \(p\equiv 1\pmod 8\) iff \(p=x^2+32y^2\). The author establishes a much more general result of this form and likewise he characterizes when \(\varepsilon_d\) is a quartic residue modulo \(p\).
On p. 418 of his book “Reciprocity laws: From Euler to Eisenstein” (2000; Zbl 0949.11002), F. Lemmermeyer proposed the problem of determining \(\varepsilon_d^{(p+1)/4}\) modulo \(p\) in terms of binary quadratic forms. The author solves this problem.
For integers \(a\) and \(b\) the Lucas sequence \(\{u_n(a,b)\}\) is defined as follows: \(u_0(a,b)=0,~u_1(a,b)=1\) and, for \(n\geq 1\), \(u_{n+1}(a,b)=bu_n(a,b)-au_{n-1}(b)\). Let \(P_n=u_n(-1,2)\) be the Pell sequence. In 1974 Emma Lehmer [J. Reine Angew. Math. 268–269, 294–301 (1974; Zbl 0289.12007)] showed that \(p\) divides the \((p-1)/4\)th terms of the Pell sequence iff \(p=x^2+32y^2\) for some integers \(x\) and \(y\). The author characterizes the primes \(p\) dividing the \((p-(-1/p))/4\)th term of the Lucas sequence \(u_n(a,b)\) in terms of representability by binary quadratic forms. This generalizes Lehmer’s result and several similar results.
The results of the author concerning the above problems generalize a large body of hitherto isolated results, but are unfortunately too complicated to be stated in this review. The proofs are quite computational and only involve a modest amount of conceptual machinery.

MSC:

11A15 Power residues, reciprocity
11E25 Sums of squares and representations by other particular quadratic forms
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
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References:

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