An efficient algorithmic solution of the Diophantine equation 𝑢²+5𝑣²=𝑚

Wilker, Peter

doi:10.1090/S0025-5718-1980-0583512-5

An efficient algorithmic solution of the Diophantine equation $u^{2}+5v^{2}=m$
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Math. Comp. 35 (1980), 1347-1352 Request permission

Abstract:

The determination of irreducible elements of the domain $Z[\sqrt { - 5} ]$ requires the solution of the Diophantine equation ${u^2} + 5{v^2} = m$, where m represents certain primes or products of two primes. An algorithm of order $\log m$ is given for the solution of the equation.

References

John Brillhart, Note on representing a prime as a sum of two squares, Math. Comp. 26 (1972), 1011–1013. MR 314745, DOI 10.1090/S0025-5718-1972-0314745-6
D. H. Lehmer, Computer technology applied to the theory of numbers, Studies in Number Theory, Math. Assoc. America, Buffalo, N.Y.; distributed by Prentice-Hall, Englewood Cliffs, N.J., 1969, pp. 117–151. MR 0246815
L. J. Mordell, Diophantine equations, Pure and Applied Mathematics, Vol. 30, Academic Press, London-New York, 1969. MR 0249355
H. M. Stark, On complex quadratic fields wth class-number two, Math. Comp. 29 (1975), 289–302. MR 369313, DOI 10.1090/S0025-5718-1975-0369313-X