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

Author: J. Fabrykowski
Journal: Proc. Amer. Math. Soc. 122 (1994), 325-331
MSC: Primary 11A07; Secondary 11B75
DOI: https://doi.org/10.1090/S0002-9939-1994-1205491-1
MathSciNet review: 1205491
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Abstract: Let , 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.

[1] B. Bollobás, Random graphs, Academic Press, New York, 1985. MR 809996 (87f:05152)
[2] D. A. Buell and K. S. Williams, Maximal residue difference sets modulo p, Proc. Amer. Math. Soc. 69 (1978), 205-209. MR 0498345 (58:16478)
[3] P. Erdös and G. Szekeres, A combinatorial problem in geometry, Compositio Math. 2 (1935), 435-470.
[4] J. Fabrykowski, On maximal residue difference sets modulo p, Canad. Math. Bull. 36 (1993), 144-146. MR 1222527 (94b:11006)
[5] R. Guy, Unsolved problems in number theory, Vol. 1, Problem F8, Springer-Verlag, New York, 1981, p. 135. MR 656313 (83k:10002)
[6] H. L. Montgomery, Topics in multiplicative number theory, Springer-Verlag, New York, 1986. MR 0337847 (49:2616)
[7] E. J. F. Primrose, The number of quadratic residues $\bmod\;m$ , Math. Gaz. 61 (1977), 60-61. MR 0460223 (57:218)
[8] I. Z. Ruzsa, Difference sets without squares, Period. Math. Hungar. 15 (1984), 205-209. MR 756185 (85j:11022)
[9] -, private communication.
[10] A. Sarközy, On difference sets of sequences of integers. II, Ann. Univ. Sci. Budapest 21 (1978), 45-53. MR 536201 (80j:10062a)