On the Diophantine Equation Mp^x + (Mq + 1)^y = z^2

In this paper, we study and solve the exponential Diophantine equation of the formMxp + (Mq + 1)y = z2 for Mersenne primes Mp and Mq and non-negative integers x, y, and z. We use elementary methods, such as the factoring method and the modular arithmetic method, to prove our research results. Several illustrations are presented, as well as cases where solutions to the Diophantine equation do not exist.

Download Full-text

ON THE EXPONENTIAL DIOPHANTINE EQUATION

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972713000956 ◽

2014 ◽

Vol 90 (1) ◽

pp. 9-19 ◽

Cited By ~ 3

Author(s):

TAKAFUMI MIYAZAKI ◽

NOBUHIRO TERAI

Keyword(s):

Positive Integer ◽

Diophantine Equation ◽

Integer Solution ◽

Exponential Diophantine Equation ◽

Positive Integer Solution ◽

Baker’S Method ◽

Elementary Methods ◽

Positive Integers

AbstractLet $m$, $a$, $c$ be positive integers with $a\equiv 3, 5~({\rm mod} \hspace{0.334em} 8)$. We show that when $1+ c= {a}^{2} $, the exponential Diophantine equation $\mathop{({m}^{2} + 1)}\nolimits ^{x} + \mathop{(c{m}^{2} - 1)}\nolimits ^{y} = \mathop{(am)}\nolimits ^{z} $ has only the positive integer solution $(x, y, z)= (1, 1, 2)$ under the condition $m\equiv \pm 1~({\rm mod} \hspace{0.334em} a)$, except for the case $(m, a, c)= (1, 3, 8)$, where there are only two solutions: $(x, y, z)= (1, 1, 2), ~(5, 2, 4). $ In particular, when $a= 3$, the equation $\mathop{({m}^{2} + 1)}\nolimits ^{x} + \mathop{(8{m}^{2} - 1)}\nolimits ^{y} = \mathop{(3m)}\nolimits ^{z} $ has only the positive integer solution $(x, y, z)= (1, 1, 2)$, except if $m= 1$. The proof is based on elementary methods and Baker’s method.

Download Full-text

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

Mathematics ◽

10.3390/math9151813 ◽

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.

Download Full-text