We study a function field version of a classical problem concerning square-free values of polynomials evaluated at primes. We show that for a square-free polynomial $f\in \mathbb{F}_q[t][x]$, there is a limiting density as $n\to \infty$ of primes $P \in \mathbb{F}_q[t]$ of degree $n$ such that $f(P)$ is square-free.
Over the integers the analogous result is only known when all irreducible factors of $f$ have degree at most 3.
We prove a modification as well as an improvement of a result of K. Ford, D. R. Heath-Brown and S. Konyagin concerning prime avoidance of square-free numbers and perfect powers of prime numbers.
Assuming the Riemann Hypothesis we prove that the interval $[N, N + H]$ contains an integer which is a sum of a prime and two squares of primes provided that $H \ge C (\log N)^{4}$, where $C > 0$ is an effective constant.
We investigate the consecutive primes $p$ and $q$ ($p > q$) for which there exists a pair of natural numbers $(x,y)$ such that $p^x-q^y$ is a perfect square and make some conjectures.
Let $[\, \cdot\,]$ be the floor function. In this paper we show that when $1<c<\frac{3849}{3334}$, then there exist infinitely many prime numbers of the form $[n^c]$, where $n$ is square-free.
By developing the method of Wooley on the quadratic Waring-Goldbach problem, we prove that all sufficiently large even integers can be expressed as a sum of four squares of primes and 46 powers of 2.
In this paper we study quadratic forms which are universal when restricted to almost prime inputs, establishing finiteness theorems akin to the Conway--Schneeberger 15 theorem.
We show that every sufficiently large integer is a sum of a prime and two almost prime squares, and also a sum of a smooth number and two almost prime squares. The number of such representations is of the expected order of magnitude. We likewise treat representations of shifted primes p-1 as sums of two almost prime squares. The methods involve a combination of analytic, automorphic and algebraic arguments to handle representations by restricted binary quadratic forms with a high degree of uniformity.
Every odd prime number p can be written in exactly (p + 1)/2 ways as a sum ab+cd of two ordered products ab and cd such that min(a, b) > max(c, d). An easy corollary is a proof of Fermat's Theorem expressing primes in 1 + 4N as sums of two squares.
We obtain an asymptotic formula for the number of ways to represent every reduced residue class as a product of a prime and square-free integer. This may be considered as a relaxed version of a conjecture of Erdös, Odlyzko, and Sárközy.
Let $k\ge 1$ be an integer. We prove that a suitable asymptotic formula for the average number of representations of integers $n=p_{1}^{k}+p_{2}^{2}+p_{3}^{2}$, where $p_1,p_2,p_3$ are prime numbers, holds in intervals shorter than the ones previously known.
We prove that every integer $n \geq 10$ such that $n \not\equiv 1 \text{mod} 4$ can be written as the sum of the square of a prime and a square-free number. This makes explicit a theorem of Erdős that every sufficiently large integer of this type may be written in such a way. Our proof requires us to construct new explicit results for primes in arithmetic progressions. As such, we use the second author's numerical computation regarding GRH to extend the explicit bounds of Ramaré-Rumely.
We analyze the inequality $\sqrt{P_{k+1}}-\sqrt{P_{k}}<1,\ k\in\mathbb{N}$, discuss the existence of primes on arbitrary intervals $(r,s),\ r<s,\ r,s\in\mathbb{R}$, and finally address the issue of primes between squares of naturals.
We study the problem of writing Gaussian primes as the sum of two squares, both of which are interesting arithmetically, in particular, when one is the square of a prime and the other the square of an almost-prime.
Almost primes and primes that are sums of two squares plus 1
Institut Élie Cartan de Lorraine, Université de Lorraine, CNRS（洛林埃利嘉当研究所，洛林大学，法国国家科学研究中心）；Department of Mathematics, University of Illinois Urbana–Champaign（伊利诺伊大学厄巴纳-香槟分校数学系）；Max-Planck-Institut für Mathematik（马克斯·普朗克数学研究所）
Let $[\, \cdot\,]$ be the floor function. In this paper, we show that when $1<c<37/36$, then every sufficiently large positive integer $N$ can be represented in the form \begin{equation*} N=[p^c_1]+[p^c_2]+[p^c_3]\,, \end{equation*} where $p_1,p_2,p_3$ are primes close to squares.
We give a new proof that there are infinitely many primes, relying on van der Waerden's theorem for coloring the integers, and Fermat's theorem that there cannot be four squares in an arithmetic progression. We go on to discuss where else these ideas have come together in the past.
Let $c$ be fixed with $1<c<35/34$. In this paper we prove that for every sufficiently large real number $N$ and a small constant $\varepsilon>0$, the diophantine inequality \begin{equation*} |p_1^c+p_2^c+p_3^c-N|<\varepsilon \end{equation*} is solvable in primes $p_1,\,p_2,\,p_3$ near to squares.
We adopt A. J. Irving's sieve method to study the almost-prime values produced by products of irreducible polynomials evaluated at prime arguments. This generalizes the previous results of Irving and Kao, who separately examined the almost-prime values of a single irreducible polynomial evaluated at prime arguments.
Let $p$ be a prime. We define $S(p)$ the smallest number $k$ such that every positive integer is a sum of at most $k$ squares of integers that are not divisible by $p$. In this article, we prove that $S(2)=10$, $S(3)=6$, $S(5)=5$, and $S(p)=4$ for any prime $p$ greater than $5$. In particular, it is proved that every positive integer is a sum of at most four squares not divisible by $5$, except the unique positive integer $79$.
Let $\mathcal{A}'$ be the set of integers missing any three fixed digits from their decimal expansion. We produce primes in a thin sequence by proving an asymptotic formula for counting primes of the form $p = m^2 + \ell^2$, with $\ell \in \mathcal{A}'$. The proof draws on ideas from the work of Friedlander-Iwaniec on primes of the form $p = x^2+y^4$, as well as ideas from the work of Maynard on primes with restricted digits.
Let $E(N)$ denote the number of positive integers $n \le N$, with $n \equiv 4 \pmod{24}$, which cannot be represented as the sum of four squares of primes. We establish that $E(N)\ll N^{11/32}$, thus improving on an earlier result of Harman and the first author, where the exponent $7/20$ appears in place of $11/32$.
We prove upper bounds for the mean square of the remainder in the prime geodesic theorem, for every cofinite Fuchsian group, which improve on average on the best known pointwise bounds. The proof relies on the Selberg trace formula. For the modular group we prove a refined upper bound by using the Kuznetsov trace formula.
We prove that if $λ_1$, $λ_2$, $λ_3$ and $λ_4$ are non-zero real numbers, not all of the same sign, $λ_1 / λ_2$ is irrational, and $\varpi$ is any real number then, for any $\eps > 0$ the inequality $ \bigl|λ_1 p_1 + λ_2 p_2^2 + λ_3 p_3^2 + λ_4 p_4^2 + \varpi \bigr| \le \bigl(\max_j p_j \bigr)^{-1 / 18 + \eps} $ has infinitely many solution in prime variables $p_1$, ..., $p_4$
We show that there are infinitely many primes $p$ such that $p-1$ is divisible by a square $d^2 \geq p^θ$ for $θ=1/2+1/2000.$ This improves the work of Matomäki (2009) who obtained the result for $θ=1/2-\varepsilon$ (with the added constraint that $d$ is also a prime), which improved the result of Baier and Zhao (2006) with $θ=4/9-\varepsilon.$ Similarly as in the work of Matomäki, we apply Harman's sieve method to detect primes $p \equiv 1 \, (d^2)$. To break the $θ=1/2$ barrier we prove a new bilinear equidistribution estimate modulo smooth square moduli $d^2$ by using a similar argument as …
We use recent bounds on bilinear sums with modular square roots to study the distribution of solutions to congruences $x^2 \equiv p \pmod q$ with primes $p\le P$ and integers $q \le Q$. This can be considered as a combined scenario of Duke, Friedlander and Iwaniec with averaging only over the modulus $q$ and of Dunn, Kerr, Shparlinski and Zaharescu with averaging only over $p$.
When the sequences of squares of primes is coloured with $K$ colours, where $K \geq 1$ is an integer, let $s(K)$ be the smallest integer such that each sufficiently large integer can be written as a sum of no more than $s(K)$ squares of primes, all of the same colour. We show that $s(K) \ll K \exp\left(\frac{(3\log 2 + {\rm o}(1))\log K}{\log \log K}\right)$ for $K \geq 2$. This improves on $s(K) \ll_ε K^{2 +ε}$, which is the best available upper bound for $s(K)$.
We refine a result of W.P. Li and Wang on the values of the form $ λ_1p_1 + λ_2p_2^{2} + λ_3p_3^{2} + μ_1 2^{m_1} +...+ μ_s 2^{m_s}, $ where $p_1,p_2,p_3$ are prime numbers, $m_1,..., m_s$ are positive integers, $λ_1,λ_2,λ_{3}$ are nonzero real numbers, not all of the same sign,$λ_2 / λ_3$ is irrational and $λ_i/μ_i \in \Q$, for $i\in\{1,2,3\}$.
Uniformly for small $q$ and $(a,q)=1$, we obtain an estimate for the weighted number of ways a sufficiently large integer can be represented as the sum of a prime congruent to $a$ modulo $q$ and a square-free integer. Our method is based on the notion of local model developed by Ramaré and may be viewed as an abstract circle method.
Hardy and Littlewood conjectured that every sufficiently large integer is either a square or the sum of a prime and a square. Let $E(x)$ be the number of positive integers up to $x\ge4$ which does not satisfy this condition. We prove $E(x)\ll x^{1/2}(\log x)^A(\log\log x)^4$with $A=3/2$ under the Generalized Riemann Hypothesis. This is a small improvement of the previous remarks of Mikawa (1993) and Perelli-Zaccagnini (1995) which claims $A=4,3$ respectively.
In 1951, Linnik proved the existence of a constant $K$ such that every sufficiently large even number is the sum of two primes and at most $K$ powers of 2. Since then, this style of approximation has been considered for problems similar to the Goldbach conjecture. One such problem is the representation of a sufficiently large even number as a sum of four squares of primes and at most $k$ powers of two. In 2014, Zhao proved this to be true with $k = 46$. In this paper, we reduce this to $k = 31$.
We prove that suitable asymptotic formulae in short intervals hold for the problems of representing an integer as a sum of a prime square and a square, or a prime square. Such results are obtained both assuming the Riemann Hypothesis and in the unconditional case.
Let $\mathcal{R}$ be a finite set of integers satisfying appropriate local conditions. We show the existence of long clusters of primes $p$ in bounded length intervals with $p-b$ squarefree for all $b \in \mathcal{R}$. Moreover, we can enforce that the primes $p$ in our cluster satisfy any one of the following conditions: (1) $p$ lies in a short interval $[N, N+N^{\frac{7}{12}+ε}]$, (2) $p$ belongs to a given inhomogeneous Beatty sequence, (3) with $c \in (\frac{8}{9},1)$ fixed, $p^c$ lies in a prescribed interval mod $1$ of length $p^{-1+c+ε}$.
Let $Λ\left(n\right)$ be the von Mangoldt function and $r_{SP}\left(n\right)=\sum_{m_{1}+m_{2}^{2}+m_{3}^{2}=n}Λ\left(m_{1}\right)Λ\left(m_{2}\right)Λ\left(m_{3}\right)$ be the counting function for the numbers that can be written as sum of a prime and two squares of primes. Let $N$ a sufficiently large integer, let $k>3/2$ and let $M_{i}\left(N,k\right),\, i=1,\dots,4$ suitable parameters depending on $Γ(s)$. We prove that $$\sum_{n\leq N}r_{SP}\left(n\right)\frac{\left(N-n\right)^{k}}{Γ\left(k+1\right)}=M_{1}\left(N,k\right)+M_{2}\left(N,k\right)+M_{3}\left(N,k\right)+M_{4}\left(N,k\right)+O…
In this note, we study the problem of existence of sequences of consecutive 1's in the periodic part of the continued fractions expansions of square roots of primes. We prove unconditionally that, for a given $N\gg 1$, there are at least $N\log^{-3/2}N$ prime numbers $p\leq N$ such that the continued fraction expansion of $\sqrt{p}$ contains three consecutive 1's on the beginning of the periodic part. We also present results of our computations related to the considered problem and some related problems, formulate several open questions and conjectures and get some results under the assumption…
Let $η>0$ be a fixed positive number, let $N$ be a sufficiently large number. In this paper, we study the second moment of the sum of Hecke eigenvalues over primes in short intervals (whose length is $η\log N$) on average (with some weights) over the family of weight $k$ holomorphic Hecke cusp forms. We also generalize the above result to Hecke-Maass cusp forms for $SL(2,\mathbb{Z})$ and $SL(3,\mathbb{Z}).$ By applying the Hardy-Littlewood prime 2-tuples conjecture, we calculate the exact values of the mean values.
We prove a uniform estimate for sums of Hecke--Maass eigenvalues squared over primes in short intervals that can be regarded as an analogue of Hoheisel's classical prime number theorem for all real analytic cusp forms. Our argument is modelled after our treatment of Linnik's least prime number theorem for arithmetic progressions (Tata LN 72) and depends on recent works in the theory of automorphic representations. We stress that constants in the present work, including those implicit, are all universal and effectively computable, although we pay no particular attention to numerical precision. …
We provide a new proof of Hua's result that every sufficiently large integer N\equiv 5(mod\,24) can be written as the sum of the five prime squares. Hua's original proof relies on the circle method and uses results from the theory of $L$-functions. Here, we present a proof based on the transference principle first introduced by Green. Using a sieve theoretic approach similar to the work by Shao, we do not require any results related to the distributions of zeros of L- functions. The main technical difficulty of our approach lies in proving the pseudorandomness of the majorant of the characteri…
