scholarly journals On the prime factors of the number 2p-1 - 1

1968 ◽  
Vol 9 (2) ◽  
pp. 83-86 ◽  
Author(s):  
A. Rotkiewicz
Keyword(s):  
Positive Integer ◽  
Natural Number ◽  
Arithmetical Progression ◽  
Prime Factors ◽  
Positive Integers ◽  
Coprime Positive Integers

From the proof of Theorem 2 of [5] it follows that for every positive integer k there exist infinitely many primes p in the arithmetical progression ax + b (x = 0, 1, 2,…), where a and b are relatively prime positive integers, such that the number 2p−1 − 1 has at least k composite factors of the form (p − 1)x + 1. The following question arises:For any given natural number k, do there exist infinitely many primes p such that the number 2p−1 − 1 has k prime factors of the form(p − 1)x + 1 and p ≡ b (mod a), where a and b are coprime positive integers?

scholarly journals ON A VARIATION OF A CONGRUENCE OF SUBBARAO

2012 ◽  
Vol 93 (1-2) ◽  
pp. 85-90 ◽  
Author(s):  
ANDREJ DUJELLA ◽  
FLORIAN LUCA
Keyword(s):  
Positive Integer ◽  
Continued Fractions ◽  
Euler Function ◽  
Prime Factors ◽  
Finite Set ◽  
Positive Integers ◽  
Sum Of Divisors

AbstractWe study positive integers $n$ such that $n\phi (n)\equiv 2\hspace{0.167em} {\rm mod}\hspace{0.167em} \sigma (n)$, where $\phi (n)$ and $\sigma (n)$ are the Euler function and the sum of divisors function of the positive integer $n$, respectively. We give a general ineffective result showing that there are only finitely many such $n$ whose prime factors belong to a fixed finite set. When this finite set consists only of the two primes $2$ and $3$ we use continued fractions to find all such positive integers $n$.

A note on weakly prime-additive numbers

2021 ◽  
pp. 1-4
Author(s):  
Jin-Hui Fang
Keyword(s):  
Positive Integer ◽  
Prime Divisors ◽  
Prime Factors ◽  
Positive Integers

A positive integer [Formula: see text] is called weakly prime-additive if [Formula: see text] has at least two distinct prime divisors and there exist distinct prime divisors [Formula: see text] of [Formula: see text] and positive integers [Formula: see text] such that [Formula: see text]. It is easy to see that [Formula: see text]. In this paper, intrigued by De Koninck and Luca’s work, we further determine all weakly prime-additive numbers [Formula: see text] such that [Formula: see text], where [Formula: see text] are distinct odd prime factors of [Formula: see text].

MINIMAL DEGREE SEQUENCE FOR TORUS KNOTS OF TYPE (p, q)

2009 ◽  
Vol 18 (04) ◽  
pp. 485-491 ◽  
Author(s):  
PRABHAKAR MADETI ◽  
RAMA MISHRA
Keyword(s):  
Positive Integer ◽  
Degree Sequence ◽  
Minimal Degree ◽  
Torus Knots ◽  
Torus Knot ◽  
Positive Integers ◽  
Coprime Positive Integers

In this paper we prove the following result: for coprime positive integers p and q with p < q, if r is the least positive integer such that 2p-1 and q + r are coprime, then the minimal degree sequence for a torus knot of type (p, q) is the triple (2p - 1, q + r, d) or (q + r, 2p - 1, d) where q + r + 1 ≤ d ≤ 2q - 1.

scholarly journals A congruence involving harmonic sums modulo pαqβ

2017 ◽  
Vol 13 (05) ◽  
pp. 1083-1094 ◽  
Author(s):  
Tianxin Cai ◽  
Zhongyan Shen ◽  
Lirui Jia
Keyword(s):  
Positive Integer ◽  
Prime Power ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Prime Factors ◽  
Necessary And Sufficient ◽  
Positive Integers

In 2014, Wang and Cai established the following harmonic congruence for any odd prime [Formula: see text] and positive integer [Formula: see text], [Formula: see text] where [Formula: see text] and [Formula: see text] denote the set of positive integers which are prime to [Formula: see text]. In this paper, we obtain an unexpected congruence for distinct odd primes [Formula: see text], [Formula: see text] and positive integers [Formula: see text], [Formula: see text] and the necessary and sufficient condition for [Formula: see text] Finally, we raise a conjecture that for [Formula: see text] and odd prime power [Formula: see text], [Formula: see text], [Formula: see text] However, we fail to prove it even for [Formula: see text] with three distinct prime factors.

The Diophantine equation (m2 + n2)x + (2mn)y = (m + n)2z

2020 ◽  
Vol 16 (08) ◽  
pp. 1701-1708
Author(s):  
Xiao-Hui Yan
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Integer Solution ◽  
Exponential Diophantine Equation ◽  
Positive Integer Solution ◽  
Positive Integers ◽  
Coprime Positive Integers

For fixed coprime positive integers [Formula: see text], [Formula: see text], [Formula: see text] with [Formula: see text] and [Formula: see text], there is a conjecture that the exponential Diophantine equation [Formula: see text] has only the positive integer solution [Formula: see text] for any positive integer [Formula: see text]. This is the analogue of Jésmanowicz conjecture. In this paper, we consider the equation [Formula: see text], where [Formula: see text] are coprime positive integers, and prove that the equation has no positive integer solution if [Formula: see text] and [Formula: see text].

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.

On the prime factors of the iterates of the Ramanujan τ–function

2020 ◽  
Vol 63 (4) ◽  
pp. 1031-1047
Author(s):  
Florian Luca ◽  
Sibusiso Mabaso ◽  
Pantelimon Stănică
Keyword(s):  
Positive Integer ◽  
Prime Factors

AbstractIn this paper, for a positive integer n ≥ 1, we look at the size and prime factors of the iterates of the Ramanujan τ function applied to n.

Congruences involving alternating harmonic sums modulo pα qβ

2018 ◽  
Vol 68 (5) ◽  
pp. 975-980
Author(s):  
Zhongyan Shen ◽  
Tianxin Cai
Keyword(s):  
Positive Integer ◽  
Positive Integers ◽  
Mod P

Abstract In 2014, Wang and Cai established the following harmonic congruence for any odd prime p and positive integer r, $$\sum_{\begin{subarray}{c}i+j+k=p^{r}\\ i,j,k\in\mathcal{P}_{p}\end{subarray}}\frac{1}{ijk}\equiv-2p^{r-1}B_{p-3} \quad\quad(\text{mod} \,\, {p^{r}}),$$ where $ \mathcal{P}_{n} $ denote the set of positive integers which are prime to n. In this note, we obtain the congruences for distinct odd primes p, q and positive integers α, β, $$ \sum_{\begin{subarray}{c}i+j+k=p^{\alpha}q^{\beta}\\ i,j,k\in\mathcal{P}_{2pq}\end{subarray}}\frac{1}{ijk}\equiv\frac{7}{8}\left(2-% q\right)\left(1-\frac{1}{q^{3}}\right)p^{\alpha-1}q^{\beta-1}B_{p-3}\pmod{p^{% \alpha}} $$ and $$ \sum_{\begin{subarray}{c}i+j+k=p^{\alpha}q^{\beta}\\ i,j,k\in\mathcal{P}_{pq}\end{subarray}}\frac{(-1)^{i}}{ijk}\equiv\frac{1}{2}% \left(q-2\right)\left(1-\frac{1}{q^{3}}\right)p^{\alpha-1}q^{\beta-1}B_{p-3}% \pmod{p^{\alpha}}. $$

scholarly journals ADDITIVE BASES AND NIVEN NUMBERS

2021 ◽  
pp. 1-8
Author(s):  
CARLO SANNA
Keyword(s):  
Positive Integer ◽  
Natural Number ◽  
Riemann Hypothesis ◽  
Additive Basis

Abstract Let $g \geq 2$ be an integer. A natural number is said to be a base-g Niven number if it is divisible by the sum of its base-g digits. Assuming Hooley’s Riemann hypothesis, we prove that the set of base-g Niven numbers is an additive basis, that is, there exists a positive integer $C_g$ such that every natural number is the sum of at most $C_g$ base-g Niven numbers.

Sign in / Sign up

Export Citation Format

Share Document