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Diophantine equations and Fermat’s last theorem for multivariate (skew-)polynomials
Fermat’s Last Theorem is a famous theorem in number theory which is difficult to prove. However, it is known that the version of polynomials with one...
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Symmetry-Based Approach to the Problem of a Perfect Cuboid
A perfect cuboid is a rectangular parallelepiped in which the lengths of all edges, the lengths of all face diagonals, and also the lengths of...
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30 years of collaboration
We highlight some of the most important cornerstones of the long standing and very fruitful collaboration of the Austrian Diophantine Number Theory...
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A new conjecture on integer powers
We make a conjecture about integer powers which states that for any integer n ≥ 2, the n th power of any arbitrary integer, including zero, can be...
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A remark on Jeśmanowicz’ conjecture for the non-coprimality case
Let a, b, c be relatively prime positive integers such that a 2 + b 2 = c 2 . Jeśmanowicz’ conjecture on Pythagorean numbers states that for any...
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Galois Action on the Homology of Fermat Curves
In his paper titled “Torsion points on Fermat Jacobians, roots of circular units and relative singular homology,” Anderson determines the homology of...
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An infinite family of multiplicatively independent bases of number systems in cyclotomic number fields
Let ζ k be a k -th primitive root of unity, m ≥ø( k ) + 1 an integer and Φ k ( X ) ∈ ℤ[ X ] the k -th cyclotomic polynomial. In this paper we show that the pair...
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The Korteweg-de Vries equation and a Diophantine problem related to Bernoulli polynomials
Some Diophantine equations related to the soliton solutions of the Korteweg-de Vries equation are resolved. The main tools are the connection with...
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The method of infinite ascent applied on A 4 ± nB 3 = C 2
Each of the Diophantine equations A 4 ± nB 3 = C 2 has an infinite number of integral solutions ( A,B,C ) for any positive integer n . In this paper, we...
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On the diophantine equation x 2 + 2 a · 3 b · 11 c = y n
In this note, we find all the solutions of the Diophantine equation x 2 + 2 a · 3 b · 11 c = y n , in nonnegative integers a, b, c, x, y, n ≥ 3 with x ...