AIMS Mathematics, 2020, 5(3): 1757-1778. doi: 10.3934/math.2020119.

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The least common multiple of consecutive terms in a cubic progression

1 School of Mathematics and Computer Science, Panzhihua University, Panzhihua 617000, P. R. China
2 Mathematical College, Sichuan University, Chengdu 610064, P.R. China

Let $k$ be a positive integer and $f(x)$ a polynomial with integer coefficients. Associated to the least common multiple ${\rm lcm}_{0\le i\le k}\{f(n+i)\}$, we define the function $\mathcal{G}_{k, f}$ for all positive integers $n\in \mathbb{N}^*\setminus Z_{k, f}$ by $\mathcal{G}_{k, f}(n):=\frac{\prod_{i=0}^k |f(n+i)|}{{\rm lcm}_{0\le i\le k}\{f(n+i)\}},$ where $Z_{k,f}:=\bigcup_{i=0}^k\{n\in \mathbb{N}^*: f(n+i)=0\}.$ If $f(x)=x$, then Farhi showed in 2007 that $\mathcal{G}_{k, f}$ is periodic with $k!$ as its period. Consequently, Hong and Yang improved Farhi's period $k!$ to ${\rm lcm}(1,...,k)$. Later on, Farhi and Kane confirmed a conjecture of Hong and Yang and determined the smallest period of $\mathcal{G}_{k, f}$. For the general linear polynomial $f(x)$, Hong and Qian showed in 2011 that $\mathcal{G}_{k, f}$ is periodic and got a formula for its smallest period. In 2015, Hong and Qian characterized the quadratic polynomial $f(x)$ such that $\mathcal{G}_{k, f}$ is almost periodic and also arrived at an explicit formula for the smallest period of $\mathcal{G}_{k, f}$. If $\deg f(x)\ge 3$, then one naturally asks the following interesting question: Is the arithmetic function $\mathcal{G}_{k,f}$ almost periodic and, if so, what is the smallest period? In this paper, we asnwer this question for the case $f(x)=x^3+2$. First of all, with the help of Hua's identity, we prove that $\mathcal{G}_{k,x^3+2}$ is periodic. Consequently, we use Hensel's lemma, develop a detailed $p$-adic analysis to $\mathcal{G}_{k, x^3+2}$ and particularly investigate arithmetic properties of the congruences $x^3+2\equiv 0 \pmod{p^e}$ and $x^6+108\equiv 0\pmod{p^e}$, and with more efforts, its smallest period is finally determined. Furthermore, an asymptotic formula for ${\rm log \ lcm}_{0 \le i \le k}\{(n+i)^3+2\}$ is given.
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Keywords cubic progression; least common multiple; p-adic valuation; almost periodic function; the smallest period; Hensel’s lemma

Citation: Zongbing Lin, Shaofang Hong. The least common multiple of consecutive terms in a cubic progression. AIMS Mathematics, 2020, 5(3): 1757-1778. doi: 10.3934/math.2020119

References

  • 1. P. Bateman, J. Kalb and A. Stenger, A limit involving least common multiples, Amer. Math. Monthly, 109 (2002), 393-394.
  • 2. P. L. Chebyshev, Memoire sur les nombres premiers, J. Math. Pures Appl., 17 (1852), 366-390.
  • 3. B. Farhi, Minorations non triviales du plus petit commun multiple de certaines suites finies d'entiers, C. R. Acad. Sci. Paris, Ser. I, 341 (2005), 469-474.    
  • 4. B. Farhi, Nontrivial lower bounds for the least common multiple of some finite sequences of integers, J. Number Theory, 125 (2007), 393-411.    
  • 5. B. Farhi, An identity involving the least common multiple of binomial coeffcients and its application, Amer. Math. Monthly, 116 (2009), 836-839.    
  • 6. B. Farhi, On the derivatives of the integer-valued polynomials, arXiv:1810.07560.
  • 7. B. Farhi and D. Kane, New results on the least common multiple of consecutive integers, Proc. Amer. Math. Soc., 137 (2009), 1933-1939.
  • 8. C. J. Goutziers, On the least common multiple of a set of integers not exceeding N, Indag. Math., 42 (1980), 163-169.
  • 9. D. Hanson, On the product of the primes, Canad. Math. Bull., 15 (1972), 33-37.    
  • 10. S. F. Hong and W. D. Feng, Lower bounds for the least common multiple of finite arithmetic progressions, C. R. Acad. Sci. Paris, Ser. I, 343 (2006), 695-698.    
  • 11. S. F. Hong, Y. Y. Luo, G. Y. Qian, et al. Uniform lower bound for the least common multiple of a polynomial sequence, C.R. Acad. Sci. Paris, Ser. I, 351 (2013), 781-785.    
  • 12. S. F. Hong and G. Y. Qian, The least common multiple of consecutive arithmetic progression terms, Proc. Edinb. Math. Soc., 54 (2011), 431-441.    
  • 13. S. F. Hong and G. Y. Qian, The least common multiple of consecutive quadratic progression terms, Forum Math., 27 (2015), 3335-3396.
  • 14. S. F. Hong and G. Y. Qian, New lower bounds for the least common multiple of polynomial sequences, J. Number Theory, 175 (2017), 191-199.    
  • 15. S. F. Hong, G. Y. Qian and Q. R. Tan, The least common multiple of a sequence of products of linear polynomials, Acta Math. Hungar., 135 (2012), 160-167.    
  • 16. S. F. Hong and Y. J. Yang, On the periodicity of an arithmetical function, C. R. Acad. Sci. Paris Sér. I, 346 (2008), 717-721.
  • 17. S. F. Hong and Y. J. Yang, Improvements of lower bounds for the least common multiple of arithmetic progressions, Proc. Amer. Math. Soc., 136 (2008), 4111-4114.    
  • 18. L.-K. Hua, Introduction to number theory, Springer-Verlag, Berlin Heidelberg, 1982.
  • 19. N. Koblitz, p-Adic numbers, p-adic analysis, and zeta-functions, Springer-Verlag, Heidelberg, 1977.
  • 20. M. Nair, On Chebyshev-type inequalities for primes, Amer. Math. Monthly, 89 (1982), 126-129.    
  • 21. J. Neukirch, Algebraic number theory, Springer-Verlag, 1999.
  • 22. S. M. Oon, Note on the lower bound of least common multiple, Abstr. Appl. Anal., 2013.
  • 23. G. Y. Qian and S. F. Hong, Asymptotic behavior of the least common multiple of consecutive arithmetic progression terms, Arch. Math., 100 (2013), 337-345.    
  • 24. G. Y. Qian, Q. R. Tan and S. F. Hong, The least common multiple of consecutive terms in a quadratic progression, Bull. Aust. Math. Soc., 86 (2012), 389-404.    
  • 25. R. J. Wu, Q. R. Tan and S. F. Hong, New lower bounds for the least common multiple of arithmetic progressions, Chinese Annals of Mathematics, Series B, 34 (2013), 861-864.    

 

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