Prime divisors of palindromes

http://www.math.missouri.edu/~bbanks/papers/index.htmlIn this paper, we study some divisibility properties of palindromic numbers in a fixed base g ≥ 2. In particular, if PL denotes the set of palindromes with precisely L digits, we show that for any sufficiently large value of L there exists a palindrome n ∈ PL with at least (log log n)1+o(1) distinct prime divisors, and there exists a palindrome n ∈ PL with a prime factor of size at least (log n)2+o(1)

Power Values of Palindromes

8 páginas.We show that for a fixed integer base g <= 2 the palindromes to base g which are k-powers form a very thin set in the set of all base g palindromes.During the preparation of this paper, J. C. was supported in part by Grant MTM 2005-04730 of MYCIT, F. L. was supported in part by Grant SEP- CONACyT 46755, and I. S. by ARC Grant DP0556431.Peer reviewe

Patterns obtained from digit and iterative digit sums of Palindromic, Repdigit and Repunit numbers, its variants and subsets

The digit and iterative digit sums of Palindromic numbers, their primes and squares, repdigit, repunit, their squares and cubes produced different patterns and sequences. The digit and iterative digits sum of the Palindromic, repdigits and repunit numbers are the same but with different pattern

Sums of Palindromes: an Approach via Automata

Recently, Cilleruelo, Luca, & Baxter proved, for all bases b >= 5, that every natural number is the sum of at most 3 natural numbers whose base-b representation is a palindrome. However, the cases b = 2, 3, 4 were left unresolved. We prove, using a decision procedure based on automata, that every natural number is the sum of at most 4 natural numbers whose base-2 representation is a palindrome. Here the constant 4 is optimal. We obtain similar results for bases 3 and 4, thus completely resolving the problem

Every positive integer is a sum of three palindromes

For integer $g\ge 5$, we prove that any positive integer can be written as a sum of three palindromes in base $g$

Reversible primes

For an $n$-bit positive integer $a$ written in binary as $$a = \sum_{j=0}^{n-1} \varepsilon_{j}(a) \,2^j$$ where, $\varepsilon_j(a) \in \{0,1\}$, $j\in\{0, \ldots, n-1\}$, $\varepsilon_{n-1}(a)=1$, let us define $$\overleftarrow{a} = \sum_{j=0}^{n-1} \varepsilon_j(a)\,2^{n-1-j},$$ the digital reversal of $a$. Also let $\mathcal{B}_n = \{2^{n-1}\leq a<2^n:~a \text{ odd}\}.$ With a sieve argument, we obtain an upper bound of the expected order of magnitude for the number of $p \in \mathcal{B}_n$ such that $p$ and $\overleftarrow{p}$ are prime. We also prove that for sufficiently large $n$, $$\left|\{a \in \mathcal{B}_n:~ \max \{\Omega (a), \Omega (\overleftarrow{a})\}\le 8 \}\right| \ge c\, \frac{2^n}{n^2},$$ where $\Omega(n)$ denotes the number of prime factors counted with multiplicity of $n$ and $c > 0$ is an absolute constant. Finally, we provide an asymptotic formula for the number of $n$-bit integers $a$ such that $a$ and $\overleftarrow{a}$ are both squarefree. Our method leads us to provide various estimates for the exponential sum $$ \sum_{a \in \mathcal{B}_n} \exp\left(2\pi i (\alpha a + \vartheta \overleftarrow{a})\right) \quad(\alpha,\vartheta \in\mathbb{R}). $

Infinitude of palindromic almost-prime numbers

It is proven that, in any given base, there are infinitely many palindromic numbers having at most six prime divisors, each relatively large. The work involves equidistribution estimates for the palindromes in residue classes to large moduli, offering upper bounds for moments and averages of certain products closely related to exponential sums over palindrome