SOLUTIONS TO A LEBESGUE–NAGELL EQUATION

2021 ◽  
pp. 1-12
Author(s):  
NGUYEN XUAN THO
Keyword(s):  
Number Theory ◽  
Computer Search ◽  
Elementary Number Theory ◽  
Lucas Sequences ◽  
Elementary Number ◽  
Primitive Divisors ◽  
Integer Solutions

Abstract We find all integer solutions to the equation $x^2+5^a\cdot 13^b\cdot 17^c=y^n$ with $a,\,b,\,c\geq 0$ , $n\geq 3$ , $x,\,y>0$ and $\gcd (x,\,y)=1$ . Our proof uses a deep result about primitive divisors of Lucas sequences in combination with elementary number theory and computer search.

An Equation Involving the Smarandache Function

2012 ◽  
Vol 204-208 ◽  
pp. 4785-4788
Author(s):  
Bin Chen
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Elementary Number Theory ◽  
Euler Function ◽  
Elementary Number ◽  
Integer Solutions

For any Positive Integer N, LetΦ(n)andS(n)Denote the Euler Function and the Smarandache Function of the Integer N.In this Paper, we Use the Elementary Number Theory Methods to Get the Solutions of the Equation Φ(n)=S(nk) if the K=9, and Give its All Positive Integer Solutions.

scholarly journals A note on the polynomial-exponential Diophantine equation (a^n − 1)(b^n − 1) = x^2

2021 ◽  
Vol 27 (3) ◽  
pp. 123-129
Author(s):  
Yasutsugu Fujita ◽  
◽  
Maohua Le ◽  
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Diophantine Equation ◽  
Exponential Diophantine Equation ◽  
Elementary Number Theory ◽  
Elementary Number ◽  
Integer Solutions ◽  
Positive Integers

For any positive integer t, let ord_2 t denote the order of 2 in the factorization of t. Let a,\,b be two distinct fixed positive integers with \min\{a,b\}>1. In this paper, using some elementary number theory methods, the existence of positive integer solutions (x,n) of the polynomial-exponential Diophantine equation (*) (a^n-1)(b^n-1)=x^2 with n>2 is discussed. We prove that if \{a,b\}\ne \{13,239\} and ord_2(a^2-1)\ne ord_2(b^2-1), then (*) has no solutions (x,n) with 2\mid n. Thus it can be seen that if \{a,b\}\equiv \{3,7\},\{3,15\},\{7,11\},\{7,15\} or \{11,15\} \pmod{16}, where \{a,b\} \equiv \{a_0,b_0\} \pmod{16} means either a \equiv a_0 \pmod{16} and b \equiv b_0\pmod{16} or a\equiv b_0 \pmod{16} and b\equiv a_0 \pmod{16}, then (*) has no solutions (x,n).

Elementary Number Theory

2016 ◽  
pp. 1-32
Author(s):  
Gary L. Mullen ◽  
James A. Sellers
Keyword(s):  
Number Theory ◽  
Elementary Number Theory ◽  
Elementary Number

Some Elementary Number Theory

2019 ◽  
pp. 239-244
Author(s):  
Richard Evan Schwartz
Keyword(s):  
Number Theory ◽  
Elementary Number Theory ◽  
Result Section ◽  
Structural Result ◽  
Elementary Number

This chapter proves some number-theoretic results about the sequences defined in Chapter 23. It proceeds as follows. Section 24.2 proves Lemma 24.1, a multipart structural result. Section 24.3 takes care of several number-theoretic details left over from Section 23.6 and Section 23.7.

Infinite Orbits Revisited

2019 ◽  
pp. 227-238
Author(s):  
Richard Evan Schwartz
Keyword(s):  
Number Theory ◽  
Elementary Number Theory ◽  
Elementary Number ◽  
Model Section

This is the first of four chapters giving a self-contained proof of Theorem 0.7. Section 23.2 describes a sequence of even rationals {pn/qn} that converges to A. Section 23.3 states the two main technical results, the Box Theorem and the Copy Theorem. Section 23.4 shows how to choose a sequence {cn}. Section 23.5 states three auxiliary results about arc copying in the plaid model. Section 23.6 deduces the Box Theorem from one of these auxiliary lemmas. Section 23.7 deduces the Copy Theorem from the auxiliary lemmas and some elementary number theory. Thus, after this chapter ends, the only remaining task is to prove the auxiliary copy lemmas and prove a few lemmas in elementary number theory.

Sign in / Sign up

Export Citation Format

Share Document