On the Ramanujan-Nagell type Diophantine equation \(Dx^2+k^n=B\)

2021 ◽  
Vol 56 (2) ◽  
pp. 263-270
Author(s):  
Zhongfeng Zhang ◽  
◽  
Alain Togbé ◽  
Keyword(s):  
Diophantine Equation ◽  
Nonnegative Integer ◽  
Integer Solutions ◽  
Positive Integers

In this paper, we prove that the Ramanujan-Nagell type Diophantine equation \(Dx^2+k^n=B\) has at most three nonnegative integer solutions \((x, n)\) for \(k\) a prime and \(B, D\) positive integers.

scholarly journals ON THE NUMBER OF SOLUTIONS OF THE DIOPHANTINE EQUATION axm−byn=c

2010 ◽  
Vol 81 (2) ◽  
pp. 177-185 ◽  
Author(s):  
BO HE ◽  
ALAIN TOGBÉ
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Number Of Solutions ◽  
Integer Solutions ◽  
Image Position ◽  
Positive Integers

AbstractLet a, b, c, x and y be positive integers. In this paper we sharpen a result of Le by showing that the Diophantine equation has at most two positive integer solutions (m,n) satisfying min (m,n)>1.

ON THE DIOPHANTINE EQUATION axy + byz + czx = 0

2012 ◽  
Vol 08 (03) ◽  
pp. 813-821 ◽  
Author(s):  
ZHONGFENG ZHANG ◽  
PINGZHI YUAN
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Integer Solutions ◽  
Positive Integers

Let a, b, c be integers. In this paper, we prove the integer solutions of the equation axy + byz + czx = 0 satisfy max {|x|, |y|, |z|} ≤ 2 max {a, b, c} when a, b, c are odd positive integers, and when a = b = 1, c = -1, the positive integer solutions of the equation satisfy max {x, y, z} < exp ( exp ( exp (5))).

scholarly journals On the Multiplicity of a Proportionally Modular Numerical Semigroup

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Ze Gu
Keyword(s):  
Positive Integer ◽  
Nonnegative Integer ◽  
Numerical Semigroup ◽  
Diophantine Inequality ◽  
Integer Solutions ◽  
Positive Integers

A proportionally modular numerical semigroup is the set S a , b , c of nonnegative integer solutions to a Diophantine inequality of the form a x   mod   b ≤ c x , where a , b , and c are positive integers. A formula for the multiplicity of S a , b , c , that is, m S a , b , c = k b / a for some positive integer k , is given by A. Moscariello. In this paper, we give a new proof of the formula and determine a better bound for k . Furthermore, we obtain k = 1 for various cases and a formula for the number of the triples a , b , c such that k ≠ 1 when the number a − c is fixed.

scholarly journals A Diophantine equation about polygonal numbers

2021 ◽  
Vol 27 (3) ◽  
pp. 113-118
Author(s):  
Yangcheng Li ◽  
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Nonnegative Integer ◽  
Diophantine Equations ◽  
Integer Solution ◽  
Pell Equation ◽  
Integer Solutions ◽  
Polygonal Numbers ◽  
Alpha Beta

It is well known that the number P_k(x)=\frac{x((k-2)(x-1)+2)}{2} is called the x-th k-gonal number, where x\geq1,k\geq3. Many Diophantine equations about polygonal numbers have been studied. By the theory of Pell equation, we show that if G(k-2)(A(p-2)a^2+2Cab+B(q-2)b^2) is a positive integer but not a perfect square, (2A(p-2)\alpha-(p-4)A + 2C\beta+2D)a + (2B(q-2)\beta-(q-4)B+2C\alpha+2E)b>0, 2G(k-2)\gamma-(k-4)G+2H>0 and the Diophantine equation \[AP_p(x)+BP_q(y)+Cxy+Dx+Ey+F=GP_k(z)+Hz\] has a nonnegative integer solution (\alpha,\beta,\gamma), then it has infinitely many positive integer solutions of the form (at + \alpha,bt + \beta,z), where p, q, k \geq 3 and p,q,k,a,b,t,A,B,G\in\mathbb{Z^+}, C,D,E,F,H\in\mathbb{Z}.

A GENERALIZATION OF A THEOREM OF BUMBY ON QUARTIC DIOPHANTINE EQUATIONS

2006 ◽  
Vol 02 (02) ◽  
pp. 195-206 ◽  
Author(s):  
MICHAEL A. BENNETT ◽  
ALAIN TOGBÉ ◽  
P. G. WALSH
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Diophantine Equations ◽  
Integer Solutions ◽  
Positive Integers

Bumby proved that the only positive integer solutions to the quartic Diophantine equation 3X4 - 2Y2 = 1 are (X, Y) = (1, 1),(3, 11). In this paper, we use Thue's hypergeometric method to prove that, for each integer m ≥ 1, the only positive integers solutions to the Diophantine equation (m2 + m + 1)X4 - (m2 + m)Y2 = 1 are (X,Y) = (1, 1),(2m + 1, 4m2 + 4m + 3).

scholarly journals A hypothetical upper bound on the heights of the solutions of a Diophantine equation with a finite number of solutions

2018 ◽  
Vol 8 (1) ◽  
pp. 109-114
Author(s):  
Apoloniusz Tyszka
Keyword(s):  
Positive Integer ◽  
Finite Number ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Rational Solution ◽  
Computable Function ◽  
Rational Solutions ◽  
Number Of Solutions ◽  
Integer Solutions ◽  
Positive Integers

Abstract We define a computable function f from positive integers to positive integers. We formulate a hypothesis which states that if a system S of equations of the forms xi· xj = xk and xi + 1 = xi has only finitely many solutions in non-negative integers x1, . . . , xi, then the solutions of S are bounded from above by f (2n). We prove the following: (1) the hypothesis implies that there exists an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions, if the solution set is finite; (2) the hypothesis implies that the question of whether or not a given Diophantine equation has only finitely many rational solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has a rational solution; (3) the hypothesis implies that the question of whether or not a given Diophantine equation has only finitely many integer solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has an integer solution; (4) the hypothesis implies that if a set M ⊆ N has a finite-fold Diophantine representation, thenMis computable.

On Equal Products of Consecutive Integers

1970 ◽  
Vol 13 (2) ◽  
pp. 255-259 ◽  
Author(s):  
R. A. Macleod ◽  
I. Barrodale
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Generalized Equation ◽  
Algebraic Numbers ◽  
Numerical Evidence ◽  
Consecutive Integers ◽  
Integer Solutions ◽  
Image Position ◽  
Positive Integers

Using the theory of algebraic numbers, Mordell [1] has shown that the Diophantine equation1possesses only two solutions in positive integers; these are given by n = 2, m = 1, and n = 14, m = 5. We are interested in positive integer solutions to the generalized equation2and in this paper we prove for several choices of k and l that (2) has no solutions, in other cases the only solutions are given, and numerical evidence for all values of k and l for which max (k, l) ≤ 15 is also exhibited.

scholarly journals A note on the Diophantine equation $|a^x-b^y|=c$

2010 ◽  
Vol 107 (2) ◽  
pp. 161
Author(s):  
Bo He ◽  
Alain Togbé ◽  
Shichun Yang
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Integer Solution ◽  
Exponential Diophantine Equation ◽  
Positive Integer Solution ◽  
Integer Solutions ◽  
Positive Integers

Let $a,b,$ and $c$ be positive integers. We show that if $(a,b) =(N^k-1,N)$, where $N,k\geq 2$, then there is at most one positive integer solution $(x,y)$ to the exponential Diophantine equation $|a^x-b^y|=c$, unless $(N,k)=(2,2)$. Combining this with results of Bennett [3] and the first author [6], we stated all cases for which the equation $|(N^k \pm 1)^x - N^y|=c$ has more than one positive integer solutions $(x,y)$.

scholarly journals The Polynomial Part of a Restricted Partition Function Related to the Frobenius Problem

10.37236/1592 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
Matthias Beck ◽  
Ira M. Gessel ◽  
Takao Komatsu
Keyword(s):  
Partition Function ◽  
Explicit Formula ◽  
Nonnegative Integer ◽  
Frobenius Problem ◽  
Restricted Partition ◽  
Integer Solutions ◽  
Positive Integers

Given a set of positive integers $ A = \{ a_{1} , \dots , a_{n} \} $, we study the number $ p_{A} (t) $ of nonnegative integer solutions $ \left( m_{1} , \dots , m_{n} \right) $ to $ \sum_{j=1}^{n} m_{j} a_{j} = t $. We derive an explicit formula for the polynomial part of $p_A$.

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).

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