On the Diophantine equation $$\sum_{j=1}^{k}jP_j^p=P_n^q$$
Sign Up to like & getrecommendations! Published in 2020 at "Acta Mathematica Hungarica"
DOI: 10.1007/s10474-020-01043-4
Abstract: We find all the solutions of the title Diophantine equation $$P_1^p+2P_2^p + \cdots +kP_k^p=P_n^q$$ in positive integer variables $$(k, n)$$, where $$P_i$$ is the $$i^{th}$$ term of the Pell sequence if the exponents p, q… read more here.
Keywords: equation; diophantine equation; equation sum;
On the exponential Diophantine equation $$(3pm^2-1)^x+(p(p-3)m^2+1)^y=(pm)^z$$(3pm2-1)x+(p(p-3)m2+1)y=(pm)z
Sign Up to like & getrecommendations! Published in 2017 at "Periodica Mathematica Hungarica"
DOI: 10.1007/s10998-016-0162-z
Abstract: Let m be a positive integer, and let p be a prime with $$p \equiv 1~(\mathrm{mod}~4).$$p≡1(mod4). Then we show that the exponential Diophantine equation $$(3pm^2-1)^x+(p(p-3)m^2+1)^y=(pm)^z$$(3pm2-1)x+(p(p-3)m2+1)y=(pm)z has only the positive integer solution $$(x, y, z)=(1, 1,… read more here.
Keywords: 3pm 3pm2; exponential diophantine; diophantine equation; equation 3pm ... See more keywords
A Diophantine equation with the harmonic mean
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DOI: 10.1007/s10998-019-00302-4
Abstract: Let $$f\in \mathbb {Q}[x]$$ f ∈ Q [ x ] be a polynomial without multiple roots and $$\deg {f}\ge 2$$ deg f ≥ 2 . We give conditions for $$f=x^2+bx+c$$ f = x 2 +… read more here.
Keywords: diophantine equation; harmonic mean; infinitely many; equation harmonic ... See more keywords
On the diophantine equation $$ax^{3} + by + c = xyz$$ax3+by+c=xyz
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DOI: 10.1007/s13370-016-0423-2
Abstract: In this paper, we study the diophantine equation $$ax^{3} + by + c = xyz$$ax3+by+c=xyz and we produce an upper bound for the number of positive integral solutions (x, y, z) of the equation. Through this, we… read more here.
Keywords: ax3 xyz; equation xyz; diophantine equation; xyz ax3 ... See more keywords
On a class of generalized Fermat equations of signature (2,2n,3)
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DOI: 10.1016/j.jnt.2021.06.019
Abstract: Abstract We consider the Diophantine equation 7 x 2 + y 2 n = 4 z 3 . We determine all solutions to this equation for n = 2 , 3 , 4 and 5.… read more here.
Keywords: fermat equations; diophantine equation; equations signature; class generalized ... See more keywords
Connectionist based models for solving Diophantine equation
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DOI: 10.1080/09720502.2020.1712844
Abstract: Abstract Diophantine equation is an important concept while studying number theory. But there is as such no standard method or procedure to find the solution of the Diophantine equation. In this article, the numerical solution… read more here.
Keywords: diophantine equation; models solving; connectionist based; based models ... See more keywords
A New Solution to a Cubic Diophantine Equation
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DOI: 10.3390/axioms11050184
Abstract: A positive integer, which can be written as the sum of two positive cubes in two different ways, is known as a “Ramanujan number”. The most famous example is 1729=103+93=123+13, which was identified by Ramanujan… read more here.
Keywords: diophantine equation; solution cubic; new solution; cubic diophantine ... See more keywords
Research on the Number of Solutions to a Special Type of Diophantine Equation (ax−1)(by−1) = 2z2
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DOI: 10.3390/math11112497
Abstract: Let b be an odd number. By using elementary methods, we prove that: (1) When x is an odd number and y is an even number, the Diophantine equation (2x−1)(by−1)=2z2 has no positive integer solution… read more here.
Keywords: diophantine equation; number; special type; odd number ... See more keywords
On the Diophantine Equation Fn = x^a \pm x^b \pm 1 in Mersenne and Fermat Numbers
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DOI: 10.5556/j.tkjm.53.2022.3973
Abstract: In this article we investigate on the representation of Fibonacci numbers in the form x^a \pm x^b pm 1, for x in the sequence of Mersenne and Fermat numbers. read more here.
Keywords: fermat numbers; mersenne fermat; diophantine equation; equation mersenne ... See more keywords