In this paper we investigate the solutions of the equation in the title, where $\phi$ is the Euler function. We first show that it suffices to find the solutions of the above equation when $m=4$ and $x$ and $y$ are coprime positive integers. For this last equation, we show that aside from a few small solutions, all the others are in a one-to-one correspondence with the Fermat primes.

Let $A\in {\mathbb{Z}}^2\times {\mathbb{Z}}^2$ be an expanding matrix, $𝒟\subset {\mathbb{Z}}^2$ a set with $|det(A\left)\right|$ elements and define $𝒯$ via the set equation $A𝒯=𝒯+𝒟$. If the two-dimensional Lebesgue measure of $𝒯$ is positive we call $𝒯$ a self-affine plane tile. In the present paper we are concerned with topological properties of $𝒯$. We show that the fundamental group ${\pi }_1\left(𝒯\right)$ of $𝒯$ is either trivial or uncountable and provide criteria for the triviality as well as the uncountability of ${\pi }_1\left(𝒯\right)$. Furthermore, we give a short proof of the fact that the closure of each component of $\mathrm{int}\left(𝒯\right)$ is a locally...

Binary signed digit representations (BSDR’s) of integers have been studied since the 1950’s. Their study was originally motivated by multiplication and division algorithms for integers and later by arithmetics on elliptic curves. Our paper is motivated by differential cryptanalysis of hash functions. We give an upper bound for the number of BSDR’s of a given weight. Our result improves the upper bound on the number of BSDR’s with minimal weight stated by Grabner and Heuberger in On the number of...

The spectrum of a weighted Dirac comb on the Thue-Morse quasicrystal is investigated by means of the Bombieri-Taylor conjecture, for Bragg peaks, and of a new conjecture that we call Aubry-Godrèche-Luck conjecture, for the singular continuous component. The decomposition of the Fourier transform of the weighted Dirac comb is obtained in terms of tempered distributions. We show that the asymptotic arithmetics of the $p$-rarefied sums of the Thue-Morse sequence (Dumont; Goldstein, Kelly and Speer; Grabner;...

The (−β)-integers are natural generalisations of the β-integers, and thus of the integers, for negative real bases. When β is the analogue of a Parry number, we describe the structure of the set of (−β)-integers by a fixed point of an anti-morphism.

Let b ≥ 2 be a fixed positive integer. We show for a wide variety of sequences {a n}n=1∞ that for almost all n the sum of digits of a n in base b is at least c b log n, where c b is a constant depending on b and on the sequence. Our approach covers several integer sequences arising from number theory and combinatorics.

We consider digit expansions ${\sum }_{j=0}^{\ell -1}{\Phi }^j\left(d_j\right)$ with an endomorphism $\Phi$ of an Abelian group. In such a numeral system, the $w$-NAF condition (each block of $w$ consecutive digits contains at most one nonzero) is shown to minimise the Hamming weight over all expansions with the same digit set if and only if it fulfills the subadditivity condition (the sum of every two expansions of weight $1$ admits an optimal $w$-NAF).This result is then applied to imaginary quadratic bases, which are used for scalar multiplication in elliptic...

We study the palindromic complexity of infinite words $u_{\beta }$, the fixed points of the substitution over a binary alphabet, $\varphi \left(0\right)=0^{a}1$, $\varphi \left(1\right)=0^{b}1$, with $a-1\ge b\ge 1$, which are canonically associated with quadratic non-simple Parry numbers $\beta$.

We study the palindromic complexity of infinite words uβ, the fixed points of the substitution over a binary alphabet, φ(0) = 0a1, φ(1) = 0b1, with a - 1 ≥ b ≥ 1, which are canonically associated with quadratic non-simple Parry numbers β.

A simple Parry number is a real number $\beta \>1$ such that the Rényi expansion of $1$ is finite, of the form $d_{\beta }\left(1\right)=t_1\cdots t_m$. We study the palindromic structure of infinite aperiodic words $u_{\beta }$ that are the fixed point of a substitution associated with a simple Parry number $\beta$. It is shown that the word $u_{\beta }$ contains infinitely many palindromes if and only if $t_1=t_2=\cdots =t_{m-1}\ge t_m$. Numbers $\beta$ satisfying this condition are the so-called confluent Pisot numbers. If $t_m=1$ then $u_{\beta }$ is an Arnoux-Rauzy word. We show that if $\beta$ is a confluent Pisot number then...