Search ResultsShowing 1-20 of 43
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arXiv:2410.14133 (Published 2024-10-18)
On Primes $p$ such that $p-b$ Has a Large Power Factor and Few Other Divisors
Comments: 41 pages, 1 figureCategories: math.NTWe establish quantitative lower bounds for the number of primes $p$ such that, for an odd integer $b$, $p - b \leq N$ and $p - b = 2^{k(N) + m} P_k \quad \text{for some } m \geq 0$, where $P_k$ is a product of at most $k$ primes other than 2, for $k \geq 2$. These results hold under various upper bounds for $2^{k(N)}$ depending on $k $, specifically when $2^{k(N)} \leq N^a$ for different values of $a$. Our approach combines techniques from Chen's theorem, weighted sieves, and Selberg's lower bound sieve, as well as results on primes in arithmetic progressions with large power factor moduli.
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arXiv:2410.10882 (Published 2024-10-10)
Type number for orders of level (N_1,N_2)
Comments: arXiv admin note: text overlap with arXiv:2402.17443Categories: math.NTLet $N_1=p_1^{2u_1+1}...p_w^{2u_w+1}$, where the $p_i$ are distinct primes, $u_1,...,u_w$ are nonnegative integers and $w$ is an odd integer, and $N_2$ be a positive integer such that $\gcd(N_1,N_2)=1$. In this paper, we give an explicit formula for the type number, i.e. the number of isomorphism classes, of orders of level $(N_1, N_2)$. The method of proof involves the Siegel-Weil formula for ternary quadratic forms.
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arXiv:2406.00797 (Published 2024-06-02)
Infinite class field tower with small root discriminant
Comments: 7 pagesCategories: math.NTWe extend Schoof's theorem from cyclic case to any finite case and apply this to construct a class of $\mathbb Z/m\mathbb Z \rtimes \mathbb Z/\varphi(n)\mathbb Z$ extensions of $\mathbb Q$, where $m$ is either a power of $2$ or an odd integer, and $n$ be any integer such that $m$ divides $n$. As an application, we give some number fields with small root discriminant, having infinite $p$-class field tower when $p=3, 5, 7$.
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arXiv:2404.14765 (Published 2024-04-23)
On the order of magnitude of certain integer sequences
Comments: 11 pages, 1 figureCategories: math.NTLet $p$ be a prime number, and let $S$ be the numerical semigroup generated by the prime numbers not less than $p$. We compare the orders of magnitude of some invariants of $S$ with each other, e. g., the biggest atom $u$ of $S$ with $p$ itself: By Harald Helfgott (arXiv:1312.7748 [math.NT]), every odd integer $N$ greater than five can be written as the sum of three prime numbers. There is numerical evidence suggesting that the summands of $N$ always can be chosen between $\frac N6$ and $\frac N2$. This would imply that $u$ is less than $6p$.
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arXiv:2309.07723 (Published 2023-09-14)
On Salem numbers which are exceptional units
Categories: math.NTBy extending a construction due to Benedict and McMullen [2], we show that for any odd integer n and for any even integer d>n+2 there are infinitely many Salem numbers $\alpha$ of degree d such that $\alpha^n-1$ is a unit. A similar result is also proved when n runs through some classes of even integers, d>n+3 and d/2 is an odd integer.
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arXiv:2301.09263 (Published 2023-01-23)
On the solutions of $x^2= By^p+Cz^p$ and $2x^2= By^p+Cz^p$ over totally real fields
Comments: Submitted for publication; Any comments are welcome. arXiv admin note: text overlap with arXiv:2207.10930Categories: math.NTIn this article, we study the solutions of certain type over $K$ of the Diophantine equation $x^2= By^p+Cz^p$ with prime exponent $p$, where $B$ is an odd integer and $C$ is either an odd integer or $C=2^r$ for $r \in \mathbb{N}$. Further, we study the non-trivial primitive solutions of the Diophantine equation $x^2= By^p+2^rz^p$ ($r\in {1,2,4,5}$) (resp., $2x^2= By^p+2^rz^p$ with $r \in \mathbb{N}$) with prime exponent $p$, over $K$. We also present several purely local criteria of $K$.
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arXiv:2211.00412 (Published 2022-11-01)
Solutions of $x_1^2+x_2^2-x_3^2=n^2$ with small $x_3$
Comments: 29 pagesCategories: math.NTFriedlander and Iwaniec investigated integral solutions $(x_1,x_2,x_3)$ of the equation $x_1^2+x_2^2-x_3^2=D$, where $D$ is square-free and satisfies the congruence condition $D\equiv 5\bmod{8}$. They obtained an asymptotic formula for solutions with $x_3\asymp M$, where $M$ is much smaller than $\sqrt{D}$. To be precise, their condition is $M\ge D^{1/2-1/1332}$. Their analysis led them to averages of certain Weyl sums. The condition of $D$ being square-free is essential in their work. We investigate the "opposite" case when $D=n^2$ is a square of an odd integer $n$. This case is different in nature and leads to sums of Kloosterman sums. We obtain an asymptotic formula for solutions with $x_3\asymp M$, where $M\ge D^{1/2}\exp\left(-(\log D)^{1/2-\varepsilon}\right)$ for any fixed $\varepsilon>0$. It is a certain term not containing Kloosterman sums that prevents us from saving a power of $D$.
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arXiv:2206.14067 (Published 2022-06-28)
Number of solutions to $a^x + b^y = c^z$, A Shorter Version
For relatively prime integers $a$ and $b$ both greater than one and odd integer $c$, there are at most two solutions in positive integers $(x,y,z)$ to the equation $a^x + b^y = c^z$. There are an infinite number of $(a,b,c)$ giving exactly two solutions.
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arXiv:2206.02589 (Published 2022-06-06)
Proof of a conjecture involving derangements and roots of unity
Comments: 5 pagesLet $n>1$ be an odd integer. For any primitive $n$-th root $\zeta$ of unity in the complex field, via the Engenvector-eigenvalue Identity we show that $$\sum_{\tau\in D(n-1)}\mathrm{sign}(\tau)\prod_{j=1}^{n-1}\frac{1+\zeta^{j-\tau(j)}}{1-\zeta^{j-\tau(j)}} =(-1)^{\frac{n-1}{2}}\frac{((n-2)!!)^2}{n}, $$ where $D(n-1)$ is the set of all derangements of $1,\ldots,n-1$. This confirms a previous conjecture of Z.-W. Sun.
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arXiv:2204.02436 (Published 2022-04-05)
On monogenity of certain pure number fields defined by $x^{2^u\cdot 3^v\cdot 5^t}-m$
Comments: arXiv admin note: substantial text overlap with arXiv:2106.01252, arXiv:2112.01133, arXiv:2111.05899, arXiv:2106.00004; text overlap with arXiv:2203.13353Categories: math.NTLet $K = \mathbb{Q} (\alpha) $ be a pure number field generated by a root $\alpha$ of a monic irreducible polynomial $ F(x) = x^{2^u\cdot 3^v\cdot 5^t}-m$, with $ m \neq \pm 1 $ a square free rational integer, $u$, $v$ and $t$ three positive integers. In this paper, we study the monogenity of $K$. We prove that if $m\not\equiv 1\md4$, $m\not\equiv \pm 1\md9$, and $m\not\in\{\pm 1, \pm 7\}\md{25}$, then $K$ is monogenic. But if {$m\equiv 1\md{4}$} or $m\equiv 1\md9$ or $m\equiv -1\md9$ and $u=2k$ for some odd integer $k$ or $u\ge 2$ and $m\equiv 1\md{25}$ or $m\equiv -1\md{25}$ and $u=2k$ for some odd integer $k$ or $u=v=1$ and $m\equiv \pm 82\md{5^4}$, then $K$ is not monogenic.
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arXiv:2202.08390 (Published 2022-02-17)
An analogue of the Robin inequality of the second type for odd integers
Comments: Tables of numerical computations are submitted. Please check it out alsoCategories: math.NTIn this paper we give a variant of the Robin inequality which states that $\frac{\sigma(n)}{n} \leq \frac{e^\gamma}{2} \log\log n + \frac{0.7399\cdots}{\log\log n}$ for any odd integer $n \geq 3$.
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arXiv:2106.12953 (Published 2021-06-24)
On the vanishing of some mock theta functions at odd roots of unity
Comments: Accepted in Research in Number TheoryCategories: math.NTWe consider the problem of whether or not certain mock theta functions vanish at the roots of unity with an odd order. We prove for any such function $f(q)$ that there exists a constant $C>0$ such that for any odd integer $n>C$ the function $f(q)$ does vanish at the primitive $n$-th roots of unity. This leads us to conjecture that $f(q)$ does not vanish at the primitive $n$-th roots of unity for any odd positive integer $n$.
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arXiv:2011.02762 (Published 2020-11-05)
Proof of a supercongruence conjecture of (F.3) of Swisher using the WZ-method
Comments: 8 pagesCategories: math.NTFor a non-negative integer $m$, let $S(m)$ denote the sum given by $$S(m):=\sum_{n=0}^{m}\frac{(-1)^n(8n+1)}{n!^3}\left(\frac{1}{4}\right)_n^3.$$ Using the powerful WZ-method, for a prime $p\equiv 3$ $($mod $4)$ and an odd integer $r>1$, we here deduce a supercongruence relation for $S\left(\frac{p^r-3}{4}\right)$ in terms of values of $p$-adic gamma function. As a consequence, we prove one of the supercongruence conjectures of (F.3) posed by Swisher. This is the first attempt to prove supercongruences for a sum truncated at $\frac{p^r-(d-1)}{d}$ when $p^r\equiv -1$ $($mod $d)$.
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arXiv:2011.02757 (Published 2020-11-05)
Extension of a conjectural supercongruence of (G.3) of Swisher using Zeilberger's algorithm
Comments: 8 pagesCategories: math.NTUsing Zeilberger's algorithm, we here give a proof of the supercongruence $$ \sum_{n=0}^{\frac{p^r-3}{4}}(8n+1)\frac{\left(\frac{1}{4}\right)_n^4}{(1)_n^4}\equiv -p^3 \sum_{n=0}^{\frac{p^{r-2}-3}{4}}(8n+1)\frac{\left(\frac{1}{4}\right)_n^4}{(1)_n^4} ~~(\text{mod }p^{\frac{3r-1}{2}}),$$ for any odd integer $r>3$. This extends the third conjectural supercongruence of (G.3) of Swisher to modulo higher prime powers than that expected by Swisher.
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arXiv:2007.09803 (Published 2020-07-19)
On $L$-Functions of Modular Elliptic Curves and Certain $K3$ Surfaces
Comments: 18 pagesCategories: math.NTInspired by Lehmer's conjecture on the nonvanishing of the Ramanujan $\tau$-function, one may ask whether an odd integer $\alpha$ can be equal to $\tau(n)$ or any coefficient of a newform $f(z)$. Balakrishnan, Craig, Ono, and Tsai used the theory of Lucas sequences and Diophantine analysis to characterize non-admissible values of newforms of even weight $k\geq 4$. We use these methods for weight $2$ and $3$ newforms and apply our results to $L$-functions of modular elliptic curves and certain $K3$ surfaces with Picard number $\ge 19$. In particular, for the complete list of weight $3$ newforms $f_\lambda(z)=\sum a_\lambda(n)q^n$ that are $\eta$-products, and for $N_\lambda$ the conductor of some elliptic curve $E_\lambda$, we show that if $|a_\lambda(n)|<100$ is odd with $n>1$ and $(n,2N_\lambda)=1$, then \begin{align*} a_\lambda(n) \in \,& \{-5,9,\pm 11,25, \pm41, \pm 43, -45,\pm47,49, \pm53,55, \pm59, \pm61, \pm 67\}\\ & \,\,\, \cup \, \{-69,\pm 71, \pm 73,75, \pm79,\pm81, \pm 83, \pm89,\pm 93 \pm 97, 99\}. \end{align*} Assuming the Generalized Riemann Hypothesis, we can rule out a few more possibilities leaving \begin{align*} a_\lambda(n) \in \{-5,9,\pm 11,25,-45,49,55,-69,75,\pm 81,\pm 93, 99\}. \end{align*}
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arXiv:2007.06453 (Published 2020-07-08)
Proof of three conjectures on determinants related to quadratic residues
Comments: 13 pagesCategories: math.NTIn this paper we confirm three conjectures of Z.-W. Sun on determinants. We first show that any odd integer $n>3$ divides the determiant $$\left|(i^2+dj^2)\left(\frac{i^2+dj^2}n\right)\right|_{0\le i,j\le (n-1)/2},$$ where $d$ is any integer and $(\frac{\cdot}n)$ is the Jacobi symbol. Then we prove some divisibility results concerning $|(i+dj)^n|_{0\le i,j\le n-1}$ and $|(i^2+dj^2)^n|_{0\le i,j\le n-1}$, where $d\not=0$ and $n>2$ are integers. Finally, for any odd prime $p$ and integers $c$ and $d$ with $p\nmid cd$, we determine completely the Legendre symbol $(\frac{S_c(d,p)}p)$, where $S_c(d,p):=|(\frac{i^2+dj^2+c}p)|_{1\le i,j\le(p-1)/2}$.
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arXiv:2005.14466 (Published 2020-05-29)
Proof of a q-supercongruence conjectured by Guo and Schlosser
Categories: math.NTIn this paper, we confirm the following conjecture of Guo and Schlosser: for any odd integer $n>1$ and $M=(n+1)/2$ or $n-1$, $$ \sum_{k=0}^{M}[4k-1]_{q^2}[4k-1]^2\frac{(q^{-2};q^4)_k^4}{(q^4;q^4)_k^4}q^{4k}\equiv (2q+2q^{-1}-1)[n]_{q^2}^4\pmod{[n]_{q^2}^4\Phi_n(q^2)}, $$ where $[n]=[n]_q=(1-q^n)/(1-q),(a;q)_0=1,(a;q)_k=(1-a)(1-aq)\cdots(1-aq^{k-1})$ for $k\geq 1$ and $\Phi_n(q)$ denotes the $n$-th cyclotomic polynomial.
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arXiv:2005.12646 (Published 2020-05-26)
Small class number fields in the family $\mathbb{Q}(\sqrt{9m^2+4m})$
Comments: 20 pages. Comments are welcomeCategories: math.NTWe study the class number one problem for real quadratic fields $\mathbb{Q}(\sqrt{9m^2+ 4m})$, where $m$ is an odd integer. We show that for $m \equiv 1 \pmod 3$ there is only one such field with class number one and only one such field with class number two.
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arXiv:2003.10883 (Published 2020-03-24)
Some $q$-congruences arising from certain identities
Comments: 7 pagesIn this paper, by constructing some identities, we prove some $q$-analogues of some congruences. For example, for any odd integer $n>1$, we show that \begin{gather*} \sum_{k=0}^{n-1} \frac{(q^{-1};q^2)_k}{(q;q)_k} q^k \equiv (-1)^{(n+1)/2} q^{(n^2-1)/4} - (1+q)[n] \pmod{\Phi_n(q)^2},\\ \sum_{k=0}^{n-1}\frac{(q^3;q^2)_k}{(q;q)_k} q^k \equiv (-1)^{(n+1)/2} q^{(n^2-9)/4} + \frac{1+q}{q^2}[n]\pmod{\Phi_n(q)^2}, \end{gather*} where the $q$-Pochhanmmer symbol is defined by $(x;q)_0=1$ and $(x;q)_k = (1-x)(1-xq)\cdots(1-xq^{k-1})$ for $k\geq1$, the $q$-integer is defined by $[n]=1+q+\cdots+q^{n-1}$ and $\Phi_n(q)$ is the $n$-th cyclotomic polynomial. The $q$-congruences above confirm some recent conjectures of Gu and Guo.
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arXiv:2001.07825 (Published 2020-01-22)
Anticyclotomic Euler systems for unitary groups
Comments: 36 pagesCategories: math.NTLet $n \geq 1$ be an odd integer. We construct an anticyclotomic Euler system for certain cuspidal automorphic representations of unitary groups with signature $(1, 2n-1)$.