Arithmetic Progressions in the Values of a Quadratic Polynomial

We study logarithmically averaged binary correlations of bounded multiplicative functions $g_{1}$ and $g_{2}$. A breakthrough on these correlations was made by Tao, who showed that the correlation average is negligibly small whenever $g_{1}$ or $g_{2}$ does not pretend to be any twisted Dirichlet character, in the sense of the pretentious distance for multiplicative functions. We consider a wider class of real-valued multiplicative functions $g_{j}$, namely those that are uniformly distributed in arithmetic progressions to fixed moduli. Under this assumption, we obtain a discorrelation estimate, showing that the correlation of $g_{1}$ and $g_{2}$ is asymptotic to the product of their mean values. We derive several applications, first showing that the numbers of large prime factors of $n$ and $n+1$ are independent of each other with respect to logarithmic density. Secondly, we prove a logarithmic version of the conjecture of Erdős and Pomerance on two consecutive smooth numbers. Thirdly, we show that if $Q$ is cube-free and belongs to the Burgess regime $Q\leqslant x^{4-\unicode[STIX]{x1D700}}$, the logarithmic average around $x$ of the real character $\unicode[STIX]{x1D712}\hspace{0.6em}({\rm mod}\hspace{0.2em}Q)$ over the values of a reducible quadratic polynomial is small.

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The Solvability of Polynomial Pell’s Equation

Journal of Institute of Science and Technology ◽

10.3126/jist.v25i2.33749 ◽

2020 ◽

Vol 25 (2) ◽

pp. 125-132

Author(s):

Bal Bahadur Tamang ◽

Ajay Singh

Keyword(s):

Trivial Solution ◽

Continued Fraction ◽

Laurent Series ◽

Quadratic Polynomial ◽

Continued Fraction Expansion ◽

Pell’S Equation ◽

Integer Polynomials

This article attempts to describe the continued fraction expansion of ÖD viewed as a Laurent series x-1. As the behavior of the continued fraction expansion of ÖD is related to the solvability of the polynomial Pell’s equation p2-Dq2=1 where D=f2+2g is monic quadratic polynomial with deg g<deg f and the solutions p, q must be integer polynomials. It gives a non-trivial solution if and only if the continued fraction expansion of ÖD is periodic.

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On three-variable expanders over finite valuation rings

Forum Mathematicum ◽

10.1515/forum-2020-0203 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Le Quang Ham ◽

Nguyen Van The ◽

Phuc D. Tran ◽

Le Anh Vinh

Keyword(s):

Valuation Ring ◽

Product Type ◽

Quadratic Polynomial ◽

Valuation Rings

AbstractLet {\mathcal{R}} be a finite valuation ring of order {q^{r}}. In this paper, we prove that for any quadratic polynomial {f(x,y,z)\in\mathcal{R}[x,y,z]} that is of the form {axy+R(x)+S(y)+T(z)} for some one-variable polynomials {R,S,T}, we have|f(A,B,C)|\gg\min\biggl{\{}q^{r},\frac{|A||B||C|}{q^{2r-1}}\bigg{\}}for any {A,B,C\subset\mathcal{R}}. We also study the sum-product type problems over finite valuation ring {\mathcal{R}}. More precisely, we show that for any {A\subset\mathcal{R}} with {|A|\gg q^{r-\frac{1}{3}}} then {\max\{|AA|,|A^{d}+A^{d}|\}}, {\max\{|A+A|,|A^{2}+A^{2}|\}}, {\max\{|A-A|,|AA+AA|\}\gg|A|^{\frac{2}{3}}q^{\frac{r}{3}}}, and {|f(A)+A|\gg|A|^{\frac{2}{3}}q^{\frac{r}{3}}} for any one variable quadratic polynomial f.

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