Effective multiplicative representations over short intervals modulo primes: Cases C = 4 and C = 6

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Abstract

This paper investigates the problem of short-interval multiplicative representations modulo primes. For a given positive integer C, an explicit lower bound p₀(C) exists such that for every prime p ≥ p₀(C), every element of the reduced residue system modulo p can be expressed as the product modulo p of two distinct integers in the interval [1,p/C). The overall argument proceeds by contradiction combined with strong induction on prime powers. For C = 4, we construct the inductive base via case analysis and derive the explicit lower bound p ≥ 199 by combining the divisibility properties of arithmetic progressions with a lower bound of the Chebyshev ψ-function. Building on this, we formulate an offset-screening and inductive framework for a specified positive integer C, in which a successful finite screening supplies the inductive base required by the subsequent theoretical argument. We carry out the complete implementation of this framework for C = 6 and obtain the explicit lower bound p ≥ 601.

Keywords

multiplicative representation in short intervals / explicit lower bound / Chebyshev ψ-function / computer enumeration

CHANGE HISTORY

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References

[1] M. Z. Garaev, Character sums in short intervals and the multiplication table modulo a large prime, Monatsh. Math., 148 (2006), 127-138. https://doi.org/10.1007/s00605-005-0390-6

[2] M. Z. Garaev, A. A. Karatsuba, The representation of residue classes by products of small integers, Proc. Edinb. Math. Soc., 50 (2007), 363-375. https://doi.org/10.1017/S0013091504001145

[3] J. Cilleruelo, M. Z. Garaev, Congruences involving product of intervals and sets with small multiplicative doubling modulo a prime and applications, Math. Proc. Cambridge Philos. Soc., 160 (2016), 477-494. https://doi.org/10.1017/S0305004116000051

[4] I. E. Shparlinski, Modular hyperbolas, Jpn. J. Math., 7 (2012), 235-294. https://doi.org/10.1007/s11537-012-1140-8

[5] D. R. Heath-Brown, Arithmetic applications of Kloosterman sums, Nieuw Arch. Wiskunde, 1 (2000), 380-384.

[6] J. B. Rosser, L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math., 6 (1962), 64-94. https://doi.org/10.1215/ijm/1255631807