scholarly journals On the maximal length of two sequences of consecutive integers with the same prime divisors

1989 ◽  
Vol 54 (3-4) ◽  
pp. 225-236 ◽  
Author(s):  
R. Balasubramanian ◽  
T. N. Shorey ◽  
M. Waldschmidt
Keyword(s):  
Maximal Length ◽  
Prime Divisors ◽  
Consecutive Integers

On the maximal length of two sequences of integers in arithmetic progressions with the same prime divisors

1996 ◽  
Vol 121 (4) ◽  
pp. 295-307 ◽  
Author(s):  
R. Balasubramanian ◽  
M. Langevin ◽  
T. N. Shorey ◽  
M. Waldschmidt
Keyword(s):  
Arithmetic Progressions ◽  
Maximal Length ◽  
Prime Divisors

scholarly journals VALUE PATTERNS OF MULTIPLICATIVE FUNCTIONS AND RELATED SEQUENCES

2019 ◽  
Vol 7 ◽  
Author(s):  
TERENCE TAO ◽  
JONI TERÄVÄINEN
Keyword(s):  
Critical Case ◽  
Indicator Function ◽  
Multiplicative Functions ◽  
Local Distribution ◽  
Stable Sets ◽  
Prime Divisors ◽  
Short Intervals ◽  
Sign Patterns ◽  
Upper Density ◽  
Consecutive Integers

We study the existence of various sign and value patterns in sequences defined by multiplicative functions or related objects. For any set $A$ whose indicator function is ‘approximately multiplicative’ and uniformly distributed on short intervals in a suitable sense, we show that the density of the pattern $n+1\in A$ , $n+2\in A$ , $n+3\in A$ is positive as long as $A$ has density greater than $\frac{1}{3}$ . Using an inverse theorem for sumsets and some tools from ergodic theory, we also provide a theorem that deals with the critical case of $A$ having density exactly $\frac{1}{3}$ , below which one would need nontrivial information on the local distribution of $A$ in Bohr sets to proceed. We apply our results first to answer in a stronger form a question of Erdős and Pomerance on the relative orderings of the largest prime factors $P^{+}(n)$ , $P^{+}(n+1),P^{+}(n+2)$ of three consecutive integers. Second, we show that the tuple $(\unicode[STIX]{x1D714}(n+1),\unicode[STIX]{x1D714}(n+2),\unicode[STIX]{x1D714}(n+3))~(\text{mod}~3)$ takes all the $27$ possible patterns in $(\mathbb{Z}/3\mathbb{Z})^{3}$ with positive lower density, with $\unicode[STIX]{x1D714}(n)$ being the number of distinct prime divisors. We also prove a theorem concerning longer patterns $n+i\in A_{i}$ , $i=1,\ldots ,k$ in approximately multiplicative sets $A_{i}$ having large enough densities, generalizing some results of Hildebrand on his ‘stable sets conjecture’. Finally, we consider the sign patterns of the Liouville function $\unicode[STIX]{x1D706}$ and show that there are at least $24$ patterns of length $5$ that occur with positive upper density. In all the proofs, we make extensive use of recent ideas concerning correlations of multiplicative functions.

scholarly journals On the order of magnitude of Jacobsthal's function

1977 ◽  
Vol 20 (4) ◽  
pp. 329-331 ◽  
Author(s):  
R. C. Vaughan
Keyword(s):  
Prime Divisors ◽  
Order Of Magnitude ◽  
Consecutive Integers ◽  
Image Position

Let n be an integer with n > 1. Jacobsthal (3) defines g(n) to be the least integer so that amongst any g(n) consecutive integers a + 1, a + 2, … a + g(n) there is at least one coprime with n. In other words, ifthenIt is probably true thatwhere ω(n) denotes the number of different prime divisors of n, and Erdos (1) has pointed out that by the small sieve it is possible to show that there is a constant C such that

scholarly journals Number of prime divisors in a product of consecutive integers

2004 ◽  
Vol 113 (4) ◽  
pp. 327-341 ◽  
Author(s):  
Shanta Laishram ◽  
T. N. Shorey
Keyword(s):  
Prime Divisors ◽  
Consecutive Integers ◽  
Number Of Prime Divisors

scholarly journals On the Ternary Exponential Diophantine Equation Equating a Perfect Power and Sum of Products of Consecutive Integers

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1813
Author(s):  
S. Subburam ◽  
Lewis Nkenyereye ◽  
N. Anbazhagan ◽  
S. Amutha ◽  
M. Kameswari ◽  
...  
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Exponential Diophantine Equation ◽  
Research Article ◽  
All Solutions ◽  
Sum Of Products ◽  
Consecutive Integers ◽  
Sums Of Products ◽  
Positive Integers

Consider the Diophantine equation yn=x+x(x+1)+⋯+x(x+1)⋯(x+k), where x, y, n, and k are integers. In 2016, a research article, entitled – ’power values of sums of products of consecutive integers’, primarily proved the inequality n= 19,736 to obtain all solutions (x,y,n) of the equation for the fixed positive integers k≤10. In this paper, we improve the bound as n≤ 10,000 for the same case k≤10, and for any fixed general positive integer k, we give an upper bound depending only on k for n.

scholarly journals A new characterization of L2(p2)

2020 ◽  
Vol 18 (1) ◽  
pp. 907-915
Author(s):  
Zhongbi Wang ◽  
Chao Qin ◽  
Heng Lv ◽  
Yanxiong Yan ◽  
Guiyun Chen
Keyword(s):  
Positive Integer ◽  
Irreducible Character ◽  
Character Degrees ◽  
Prime Divisors

Abstract For a positive integer n and a prime p, let {n}_{p} denote the p-part of n. Let G be a group, \text{cd}(G) the set of all irreducible character degrees of G , \rho (G) the set of all prime divisors of integers in \text{cd}(G) , V(G)=\left\{{p}^{{e}_{p}(G)}|p\in \rho (G)\right\} , where {p}^{{e}_{p}(G)}=\hspace{.25em}\max \hspace{.25em}\{\chi {(1)}_{p}|\chi \in \text{Irr}(G)\}. In this article, it is proved that G\cong {L}_{2}({p}^{2}) if and only if |G|=|{L}_{2}({p}^{2})| and V(G)=V({L}_{2}({p}^{2})) .

On triplets with descending largest prime factors

2001 ◽  
Vol 38 (1-4) ◽  
pp. 45-50 ◽  
Author(s):  
A. Balog
Keyword(s):  
Prime Factor ◽  
Prime Factors ◽  
Consecutive Integers

For an integer n≯1 letP(n) be the largest prime factor of n. We prove that there are infinitely many triplets of consecutive integers with descending largest prime factors, that is P(n - 1) ≯P(n)≯P(n+1) occurs for infinitely many integers n.

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