On a Discriminator for the Polynomial $$f(x)=x^3+x$$
Communications in Mathematics and Statistics - Trang 1-14 - 2023
Tóm tắt
Let
$$\Delta (n)$$
denote the smallest positive integer m such that
$$a^3+a(1\leqslant a\leqslant n)$$
are pairwise distinct modulo m. The purpose of this paper is to determine
$$\Delta (n)$$
for all positive integers n.
Tài liệu tham khảo
Arnold, L.K., Benkoski, S.J., McCabe, B.J.: The discriminator (a simple application of Bertrand’s postulate). Am. Math. Mon. 92, 275–277 (1985)
Bremser, P.S., Schumer, P.D., Washinyton, L.C.: A note on the incongruence of consecutive integers to a fixed power. J. Number Theory 35, 105–108 (1990)
Iwaniec, H.: Topic in Classical Automorphic Forms. American Mathematical Society, Providence (1997)
Moree, P.: The incongruence of consecutive values of polynomials. Finite Fields Appl. 2, 321–335 (1996)
Moree, P., Mullen, G.L.: Dickson polynomial discriminators. J. Number Theory 59, 88–105 (1996)
Schumer, P.D.: On the incongruence of consecutive cubes. Math. Stud. 58, 42–48 (1990)
Sun, Z.-W.: On functions taking only prime values. J. Number Theory 133, 2794–2812 (2013)
Sun, Z.-W.: New Conjectures in Number Theory and Combinatorics. Harbin Institute of Technology Press, Harbin (2021)
Yang, Q.-H., Zhao, L.: On a conjecture of Sun involving powers of three. arXiv:2111.02746
Zieve, M.: A note on discriminator. J. Number Theory 73, 122–138 (1998)