On quadratic residues and nonresidues in difference sets modulo 𝑚

Fabrykowski, J.

doi:10.1090/S0002-9939-1994-1205491-1

On quadratic residues and nonresidues in difference sets modulo $m$
HTML articles powered by AMS MathViewer

by J. Fabrykowski PDF

Proc. Amer. Math. Soc. 122 (1994), 325-331 Request permission

Abstract:

Let $m > 1$, and consider a set $\mathcal {A} = \{ {a_i}\}$ of residues modulo m such that ${a_i}$ and ${a_i} - {a_j}$ for all i and j with $i \ne j$ are quadratic residues (nonresidues) modulo m. We investigate the estimation of the maximal cardinality of such a set $\mathcal {A}$ for various moduli m.

References

Béla Bollobás, Random graphs, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1985. MR 809996
Duncan A. Buell and Kenneth S. Williams, Maximal residue difference sets modulo $p$, Proc. Amer. Math. Soc. 69 (1978), no. 2, 205–209. MR 498345, DOI 10.1090/S0002-9939-1978-0498345-0

A combinatorial problem in geometry

2

J. Fabrykowski, On maximal residue difference sets modulo $p$, Canad. Math. Bull. 36 (1993), no. 2, 144–146. MR 1222527, DOI 10.4153/CMB-1993-021-7
Richard K. Guy, Unsolved problems in number theory, Problem Books in Mathematics, Springer-Verlag, New York-Berlin, 1981. MR 656313
Hugh L. Montgomery, Topics in multiplicative number theory, Lecture Notes in Mathematics, Vol. 227, Springer-Verlag, Berlin-New York, 1971. MR 0337847, DOI 10.1007/BFb0060851
E. J. F. Primrose, The number of quadratic residues $\textrm {mod}\ m$, Math. Gaz. 61 (1977), no. 415, 60–61. MR 460223, DOI 10.2307/3617445
I. Z. Ruzsa, Difference sets without squares, Period. Math. Hungar. 15 (1984), no. 3, 205–209. MR 756185, DOI 10.1007/BF02454169
A. Sárközy, On difference sets of sequences of integers. II, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 21 (1978), 45–53 (1979). MR 536201