Takafumi Miyazaki
Gunma University
CombinatoricsDiscrete mathematicsAlgebraabc conjectureSquare (algebra)Formulas for generating Pythagorean triplesExponential functionNumber theorySquare numberGeneralizationTree of primitive Pythagorean triplesPure mathematicsDiophantine equationCoprime integersCongruence (manifolds)ConjecturePrime factorMathematicsPythagorean fieldDiophantine setIntegerPythagorean tripleFermat's Last Theorem
26Publications
9H-index
173Citations
Publications 25
#1Takafumi Miyazaki (Gunma University)H-Index: 9
#2Masaki Sudo (Seikei University)
Last. Nobuhiro Terai (Oita University)H-Index: 6
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For any fixed integers a and b greater than 1, we study the Diophantine equation null null null null null null null null null null null null null null null null null null null null null null null null null null null null null null null null null null null null null null null null null null null null null null null null null null null null null null null null null null \$a^x+(ab+1)^y=b^z null null null null null null null null null null null null null null null null null null null null null nul...
#1Takafumi MiyazakiH-Index: 9
#2István PinkH-Index: 9
For any fixed coprime positive integers a,band cwith \min\{a,b,c\}>1 we prove that the equation a^x+b^y=c^zhas at most two solutions in positive integers x,yand z except for one specific case. Our result is essentially sharp in the sense that there are infinitely many examples allowing the equation to have two solutions in positive integers. From the viewpoint of a well-known generalization of Fermat's equation, it is also regarded as a 3-variable generalization of the celebrat...
A set of positive integers is called a Diophantine tuple if the product of any two elements in the set increased by 1 is a perfect square. A conjecture in this field asserts that any Diophantine triple can be uniquely extended to a Diophantine quadruple in some sense. This problem is reduced to study the coincidence between certain two binary recurrent sequences of integers. As an analogy of this, we consider a similar coincidence on the polynomial ring in one variable over the integers, and det...
#1Takafumi MiyazakiH-Index: 9
#2Nobuhiro TeraiH-Index: 6
#1Takafumi Miyazaki (Gunma University)H-Index: 9
For any given odd prime p and a fixed positive integer D prime to p, we study the equation \(x^2+D^m=p^n\) in positive integers x, m and n. We use a classical work of Dem’janenko in 1965 on a certain quadratic Diophantine equation together with some results concerning the existence of primitive divisors of Lucas sequences to examine our equation when D is a product of \(p-1\) and a square.
#1Takafumi Miyazaki (Gunma University)H-Index: 9
#1Elif KızıldereH-Index: 1
#2Takafumi MiyazakiH-Index: 9
Last. Gökhan SoydanH-Index: 5
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#2Yasutsugu Fujita (College of Industrial Technology)H-Index: 11
Last. Takafumi Miyazaki (Gunma University)H-Index: 9
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A set of positive integers is called a Diophantine tuple if the product of any two elements in the set increased by unity is a perfect square. Any Diophantine triple is conjectured to be uniquely extended to a Diophantine quadruple by joining an element exceeding the maximal element in the triple. A previous work of the second and third authors revealed that the number of such extensions for a fixed Diophantine triple is at most 11. In this paper, we show that the number is at most eight.
#1Yasutsugu Fujita (College of Industrial Technology)H-Index: 11
#2Takafumi Miyazaki (Gunma University)H-Index: 9
#1Attila Bérczes (University of Debrecen)H-Index: 10
#2Lajos Hajdu (University of Debrecen)H-Index: 14
Last. István Pink (University of Debrecen)H-Index: 9
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Abstract We provide all solutions of the title equation in positive integers x , k , y , n with 1 ≤ x 25 and n ≥ 3 . For these values of the parameters, our result gives an affirmative answer to a related, classical conjecture of Schaffer. In our proofs we combine several tools: Baker's method (in particular, sharp bounds for the linear combinations of logarithms of two algebraic numbers), polynomial-exponential congruences and computational methods.