相关论文: Second numbers in arithmetic progressions
In this paper, we improve the error term in a previous paper on an asymptotic formula for the number of squarefull numbers in an arithmetic progression.
The conclusion that the length of an arithmetic progression of the form ${3^x+2^y}$ is at most six is proved.
In this paper we continue the investigations about unlike powers in arithmetic progression. We provide sharp upper bounds for the length of primitive non-constant arithmetic progressions consisting of squares/cubes and $n$-th powers.
In the present paper we prove that there exist infinitely many arithmetic progressions of three different primes $p_1,p_2,p_3=2p_2-p_1$ such that $p_1=x_1^2 + y_1^2 +1$, $p_2=x_2^2 + y_2^2 +1$.
We introduce and study expansions of real numbers with respect to two integer bases.
For an integer $b\geq 2$, we call a positive integer $b$-anti-Niven if it is relatively prime to the sum of the digits in its base-$b$ representation. In this article, we investigate the maximum lengths of arithmetic progressions of…
We show that if besides the primes some other sequences (involving the Liouville function and the primes) have a common distribution level exceeding 0.7231 then for any positive even integer $h$ there are arbitrarily long arithmetic…
Let $s_2$ be the sum-of-digits function in base $2$, which returns the number of non-zero binary digits of a nonnegative integer $n$. We study $s_2$ alon g arithmetic subsequences and show that --- up to a shift --- the set of $m$-tuples of…
Green and Tao proved that the primes contains arbitrarily long arithmetic progressions. We show that, essentially the same proof leads to the following result: The primes in an short interval contains many arithmetic progressions of any…
Permutations of the positive integers avoiding arithmetic progressions of length $5$ were constructed in (Davis et al, 1977), implying the existence of permutations of the integers avoiding arithmetic progressions of length $7$. We…
In this paper, we study $k$-term arithmetic progressions $N, N+d, ..., N+(k-1)d$ of powerful numbers. Under the $abc$-conjecture, we obtain $d \gg_\epsilon N^{1/2 - \epsilon}$. On the other hand, there exist infinitely many $3$-term…
We develop a theory of multiplicative functions (with values inside or on the unit circle) in arithmetic progressions analogous to the well-known theory of primes in arithmetic progressions.
An overview of the results of new exhaustive computations of gaps between primes in arithmetic progressions is presented. We also give new numerical results for exceptionally large least primes in arithmetic progressions.
We prove an upper bound for the length of an arithmetic progression represented by an irreducible integral binary quadratic form or a norm form, which depends only on the form and the progression's common difference. For quadratic forms,…
If $a$ and $b$ are integers with $b>a>1$, we completely characterize ``long'' arithmetic progressions in the sumsets of the geometric progressions $1, a, a^2, a^3, \ldots$ and $1, b, b^2, b^3, \ldots$. Our proofs utilize recent applications…
For an integer $b\geq 2$, a positive integer is called a $b$-Niven number if it is a multiple of the sum of the digits in its base-$b$ representation. In this article, we show that every arithmetic progression contains infinitely many…
In this paper we collect some results about arithmetic progressions of higher order, also called polynomial sequences. Those results are applied to $(m,q)$-isometric maps.
Work in progress concerning alternative formalizations of arithmetic.
The main result of the paper is that assuming that the level $\theta$ of distribution of primes exceeds 1/2, then there exists a positive $d\leq C(\theta)$ such that there are arbitrarily long arithmetic progressions with the property that…
In this paper, we describe the structure of finite groups whose element orders or proper (abelian) subgroup orders form an arithmetic progression of ratio $r\geq 2$. This extends the case $r=1$ studied in previous papers \cite{1,8,4}.