scholarly journals On Fixed Points of Iterations Between the Order of Appearance and the Euler Totient Function

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1796
Author(s):  
Štěpán Hubálovský ◽  
Eva Trojovská
Keyword(s):  
Positive Integer ◽  
Fixed Points ◽  
Natural Number ◽  
Positive Solutions ◽  
Diophantine Equation ◽  
Fibonacci Number ◽  
Euler Totient Function

Let Fn be the nth Fibonacci number. The order of appearance z(n) of a natural number n is defined as the smallest positive integer k such that Fk≡0(modn). In this paper, we shall find all positive solutions of the Diophantine equation z(φ(n))=n, where φ is the Euler totient function.

THE FIBONACCI-NORM OF A POSITIVE INTEGER: OBSERVATIONS AND CONJECTURES

2010 ◽  
Vol 06 (02) ◽  
pp. 371-385 ◽  
Author(s):  
JEONG SOON HAN ◽  
HEE SIK KIM ◽  
J. NEGGERS
Keyword(s):  
Positive Integer ◽  
Natural Number ◽  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Natural Numbers

In this paper, we define the Fibonacci-norm [Formula: see text] of a natural number n to be the smallest integer k such that n|Fk, the kth Fibonacci number. We show that [Formula: see text] for m ≥ 5. Thus by analogy we say that a natural number n ≥ 5 is a local-Fibonacci-number whenever [Formula: see text]. We offer several conjectures concerning the distribution of local-Fibonacci-numbers. We show that [Formula: see text], where [Formula: see text] provided Fm+k ≡ Fm (mod n) for all natural numbers m, with k ≥ 1 the smallest natural number for which this is true.

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 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.

scholarly journals On the integers represented by x4 − y4

2007 ◽  
Vol 76 (1) ◽  
pp. 133-136 ◽  
Author(s):  
Andrzej Dąbrowski
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Diophantine Equation ◽  
Trivial Solution ◽  
Effective Constant

Let p be a prime number ≥ 5, and n a positive integer > 1. This note is concerned with the diophantine equation x4 − y4 = nzp. We prove that, under certain conditions on n, this equation has no non-trivial solution in Z if p ≥ C(n), where C(n) is an effective constant.

scholarly journals Solution of certain Pell equations

2018 ◽  
Vol 11 (04) ◽  
pp. 1850056 ◽  
Author(s):  
Zahid Raza ◽  
Hafsa Masood Malik
Keyword(s):  
Positive Integer ◽  
Fundamental Solution ◽  
Positive Solutions ◽  
Continued Fraction ◽  
Pell Equation ◽  
Lucas Sequences ◽  
Pell Equations ◽  
Integer Solutions ◽  
Positive Integers

Let [Formula: see text] be any positive integers such that [Formula: see text] and [Formula: see text] is a square free positive integer of the form [Formula: see text] where [Formula: see text] and [Formula: see text] The main focus of this paper is to find the fundamental solution of the equation [Formula: see text] with the help of the continued fraction of [Formula: see text] We also obtain all the positive solutions of the equations [Formula: see text] and [Formula: see text] by means of the Fibonacci and Lucas sequences.Furthermore, in this work, we derive some algebraic relations on the Pell form [Formula: see text] including cycle, proper cycle, reduction and proper automorphism of it. We also determine the integer solutions of the Pell equation [Formula: see text] in terms of [Formula: see text] We extend all the results of the papers [3, 10, 27, 37].

scholarly journals On the Diophantine Equation x2 – kxy + ky2 + ly = 0, l = 2n

2017 ◽  
Vol 55 (1) ◽  
pp. 115-118
Author(s):  
Sukrawan Mavecha
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Integer Solution ◽  
Positive Integer Solution

AbstractWe consider the Diophantine equation x2-kxy+ky2+ ly = 0 for l = 2nand determine for which values of the odd integer k, it has a positive integer solution x and y.

scholarly journals ON EXISTENCE OF POSITIVE SOLUTIONS OF NEUTRAL DIFFERENCE EQUATIONS

1994 ◽  
Vol 25 (3) ◽  
pp. 257-265
Author(s):  
J. H. SHEN ◽  
Z. C. WANG ◽  
X. Z. QIAN
Keyword(s):  
Positive Integer ◽  
Difference Equation ◽  
Positive Solution ◽  
Positive Solutions ◽  
Difference Equations ◽  
Real Numbers ◽  
Neutral Difference Equation ◽  
Neutral Difference Equations ◽  
Existence Of Positive Solutions

Consider the neutral difference equation \[\Delta(x_n- cx_{n-m})+p_nx_{n-k}=0, n\ge N\qquad (*) \] where $c$ and $p_n$ are real numbers, $k$ and $N$ are nonnegative integers, and $m$ is positive integer. We show that if \[\sum_{n=N}^\infty |p_n|<\infty \qquad (**) \] then Eq.(*) has a positive solution when $c \neq 1$. However, an interesting example is also given which shows that (**) does not imply that (*) has a positive solution when $c =1$.

scholarly journals A note on primes in certain residue classes

2018 ◽  
Vol 14 (08) ◽  
pp. 2219-2223
Author(s):  
Paolo Leonetti ◽  
Carlo Sanna
Keyword(s):  
Positive Integer ◽  
Asymptotic Density ◽  
Residue Classes ◽  
Euler Totient Function ◽  
The One ◽  
Positive Integers

Given positive integers [Formula: see text], we prove that the set of primes [Formula: see text] such that [Formula: see text] for [Formula: see text] admits asymptotic density relative to the set of all primes which is at least [Formula: see text], where [Formula: see text] is the Euler totient function. This result is similar to the one of Heilbronn and Rohrbach, which says that the set of positive integer [Formula: see text] such that [Formula: see text] for [Formula: see text] admits asymptotic density which is at least [Formula: see text].

scholarly journals A GENERALIZATION OF THE RAMANUJAN–NAGELL EQUATION

2018 ◽  
Vol 61 (03) ◽  
pp. 535-544
Author(s):  
TOMOHIRO YAMADA
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Integer Solutions

AbstractWe shall show that, for any positive integer D &gt; 0 and any primes p1, p2, the diophantine equation x2 + D = 2sp1kp2l has at most 63 integer solutions (x, k, l, s) with x, k, l ≥ 0 and s ∈ {0, 2}.

scholarly journals A NOTE ON THE DIOPHANTINE EQUATION

2013 ◽  
Vol 89 (2) ◽  
pp. 316-321 ◽  
Author(s):  
MOU JIE DENG
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Pythagorean Triple ◽  
Positive Integers

AbstractLet $(a, b, c)$ be a primitive Pythagorean triple satisfying ${a}^{2} + {b}^{2} = {c}^{2} . $ In 1956, Jeśmanowicz conjectured that for any given positive integer $n$ the only solution of $\mathop{(an)}\nolimits ^{x} + \mathop{(bn)}\nolimits ^{y} = \mathop{(cn)}\nolimits ^{z} $ in positive integers is $x= y= z= 2. $ In this paper, for the primitive Pythagorean triple $(a, b, c)= (4{k}^{2} - 1, 4k, 4{k}^{2} + 1)$ with $k= {2}^{s} $ for some positive integer $s\geq 0$, we prove the conjecture when $n\gt 1$ and certain divisibility conditions are satisfied.

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