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The number of solutions of the congruence \(y^2 \equiv x^4 - a\pmod p\). (English) Zbl 0298.10016

In this paper, the authors prove the following theorem: Let \(p\) be a prime and let \((a,p) = 1\). The number of solutions of the congruence \(y^2 \equiv x^4 - a\pmod p\) is \(p - 1\) if \(p\equiv 3\pmod 4\), and \(p - (a/\pi)_4 \bar\pi - (a/\bar\pi)_4 \pi - 1\) if \(p\equiv 1\pmod 4\). Here \((x/y)_4\) is the biquadratic residue symbol and \(p=\pi\bar\pi\) is the factorization of \(p\) into Gaussian primes, each normalized \(\equiv 1\pmod{2(1+ i)}\).
The proof depends upon results of L. E. Dickson [Am. J. Math. 57, 391–424 (1935; Zbl 0012.01203)] on cyclotomy in the case \(p\equiv 1\pmod 4\); as such, as readers of that paper (“Cyclotomy, higher congruences, and Waring’s problem”) are aware, the proof is both elementary and technically intricate. For \(p\equiv 3\pmod 4\), the authors give a straightforward proof; they also relate their result to that of B. Morlaye [Enseign. Math. (2) 18 (1972), 269–276 (1973; Zbl 0255.12007)] who proved that the number of solutions to the title congruence is one less than the number of solutions to \(y^2 \equiv x^3 - ax\pmod p\). However, Morlaye used different techniques.
Reviewer: Ezra Brown

MSC:

11D45 Counting solutions of Diophantine equations
11D79 Congruences in many variables
11T22 Cyclotomy
11A07 Congruences; primitive roots; residue systems
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