### A conjecture concerning the exponential diophantine equation ${a}^{x}+{b}^{y}={c}^{z}$

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In this note we prove that the equation $\left(\genfrac{}{}{0pt}{}{k}{2}\right)-1={q}^{n}+1$, $q\ge 2,n\ge 3$, has only finitely many positive integer solutions $(k,q,n)$. Moreover, all solutions $(k,q,n)$ satisfy $k{10}^{{10}^{182}}$, $q{10}^{{10}^{165}}$ and $n2\xb7{10}^{17}$.

The famous problem of determining all perfect powers in the Fibonacci sequence ${\left({F}_{n}\right)}_{n\ge 0}$ and in the Lucas sequence ${\left({L}_{n}\right)}_{n\ge 0}$ has recently been resolved [10]. The proofs of those results combine modular techniques from Wiles’ proof of Fermat’s Last Theorem with classical techniques from Baker’s theory and Diophantine approximation. In this paper, we solve the Diophantine equations ${L}_{n}={q}^{a}{y}^{p}$, with $a\>0$ and $p\ge 2$, for all primes $q\<1087$ and indeed for all but $13$ primes $q\<{10}^{6}$. Here the strategy of [10] is not sufficient due to the sizes of...

A generalization of the well-known Fibonacci sequence ${F\u2099}_{n\ge 0}$ given by F₀ = 0, F₁ = 1 and ${F}_{n+2}={F}_{n+1}+F\u2099$ for all n ≥ 0 is the k-generalized Fibonacci sequence ${F{\u2099}^{\left(k\right)}}_{n\ge -(k-2)}$ whose first k terms are 0,..., 0, 1 and each term afterwards is the sum of the preceding k terms. For the Fibonacci sequence the formula $F\u2099\xb2+F{\xb2}_{n+1}\xb2={F}_{2n+1}$ holds for all n ≥ 0. In this paper, we show that there is no integer x ≥ 2 such that the sum of the xth powers of two consecutive k-generalized Fibonacci numbers is again a k-generalized Fibonacci number. This generalizes...

In this paper, we look at various arithmetic properties of the set of those positive integers n whose sum of digits in a fixed base b > 1 is a fixed positive integer s. For example, we prove that such integers can have many prime factors, that they are not very smooth, and that most such integers have a large prime factor dividing the value of their Euler φ function.

The subject of the talk is the recent work of Mihăilescu, who proved that the equation ${x}^{p}-{y}^{q}=1$ has no solutions in non-zero integers $x,y$ and odd primes $p,q$. Together with the results of Lebesgue (1850) and Ko Chao (1865) this implies the celebratedconjecture of Catalan (1843): the only solution to ${x}^{u}-{y}^{v}=1$ in integers $x,y\>0$ and $u,v\>1$ is ${3}^{2}-{2}^{3}=1$. Before the work of Mihăilescu the most definitive result on Catalan’s problem was due to Tijdeman (1976), who proved that the solutions of Catalan’s equation are bounded by an absolute...

Let G be a commutative algebraic group defined over a number field K that is disjoint over K from ${}_{a}$ and satisfies the condition of semistability. Consider a linear form l on the Lie algebra of G with algebraic coefficients and an algebraic point u in a p-adic neighbourhood of the origin with the condition that l does not vanish at u. We give a lower bound for the p-adic absolute value of l(u) which depends up to an effectively computable constant only on the height of the linear form, the height...