On special Wieferich's primes

Distribution of Non-Wieferich primes in certain algebraic groups under ABC

Under ABC, Silverman showed that there are infinitely many non-Wieferich primes with respect to any (non-trivial) base $a$. Recently Srinivas and Subramani proved an analogous result over number fields with trivial class group. In the first part of this article, we extend their result to any arbitrary number fields. Secondly, we give an asymptotic lower bound for the number of non-Wieferich prime ideals. Furthermore, we show a lower bound of same order is achievable for non-Wieferich prime ideals having norm congruent to $1 \pmod k.$ Lastly, we generalize Silverman's work for elliptic curves o…

Non-Wieferich primes in number fields and ABC conjecture

Let $K/\mathbb{Q}$ be an algebraic number field of class number one and $\mathcal{O}_K$ be its ring of integers. We show that there are infinitely many non-Wieferich primes with respect to certain units in $\mathcal{O}_K$ under the assumption of the \textit{abc} conjecture for number fields.

Drinfeld 模的 Wieferich 素数

Wieferich primes for Drinfeld modules

本文在 Drinfeld 模框架下引入 Wieferich 素数，发现其与特殊 L 值 p-adic 赋值相关，推广 Thakur 定理，并证明统计分布中概率为 q^{-d}。

A p-adic identity for Wieferich primes

Let $n$ be a positive integer, $p$ be an odd prime and integers $a,b \not= 0$ with $gcd(a,b)=1$, $p \nmid ab$, and $p|(a^n \pm b^n)$, we prove the identity $$ν_p(a^n \pm b^n)-ν_p(n)=ν_p(a^{p-1}-b^{p-1}).$$ An unintended interesting immediate consequence is the following variant of Wieferich's criterion for FLT : Let $x^n+y^n=z^n$ with $n$ prime and $x,y,z$ pairwise relatively prime. Then every odd prime $p|y$ satisfies $ν_p(z^{p-1}-x^{p-1}) \ge n-1$ and every odd prime $p|x$ satisfies $ν_p(z^{p-1}-y^{p-1}) \ge n-1$, and every odd prime $p|z$ satisfies $ν_p(x^{p-1}-y^{p-1}) \ge n-1$, ie. every …

Overpseudoprimes, Mersenne Numbers and Wieferich primes

We introduce a new class of pseudoprimes-so called "overpseudoprimes" which is a special subclass of super-Poulet pseudoprimes. Denoting via h(n) the multiplicative order of 2 modulo n, we show that odd number n is overpseudoprime iff value of h(n) is invariant of all divisors d>1 of n. In particular, we prove that all composite Mersenne numbers 2^p-1,where p is prime, and squares of Wieferich primes are overpseudoprimes. We give also a generalization of the results on arbitrary base a>1 and prove that every overpseudoprime is strong pseudoprime of the same base.

广义Wieferich素数与单基因三项式

Generalized Wieferich primes and monogenic trinomials

该研究将关于x^(2p)+2x^p+2单基因性与Wieferich素数关联的结论，推广到特定条件下的多项式x^(2n)+bx^n+b，拓展了单基因三项式的判定相关成果。

Wieferich素数与单生成三项式

Wieferich Primes and Monogenic Trinomials

本文证明了多项式F_p(x)=x^{2p}+2x^p+2是单生成的当且仅当p不是Wieferich素数。

Wieferich Primes and a mod $p$ Leopoldt Conjecture

We consider questions in Galois cohomology which arise by considering mod $p$ Galois representations arising from automorphic forms. We consider a Galois cohomological analog for the standard heuristics about the distribution of Wieferich primes, i.e. prime $p$ such that $2^{p-1}$ is 1 mod $p^2$. Our analog relates to asking if in a compatible system of Galois representations, for almost all primes $p$, the residual mod $p$ representation arising from it has unobstructed deformation theory. This analog leads in particular to formulating a mod $p$ analog for almost all primes $p$ of the classic…

完全数、Wieferich素数与组合数方程$\binom{2n}{n}\boldsymbol{\text{≡}}2^n \bmod n$的解

Perfect numbers, Wieferich primes and the solutions of $\binom{2n}{n}\equiv 2^n \bmod n$

本文研究同余方程$\binom{2n}{n}\boldsymbol{\text{≡}}2^n \bmod n$的解，借助SageMath软件及自身结果增加了已知解的数量，还证明了偶完全数、Wieferich素数与该方程解的关联。

广义Wieferich素数及单生成多项式

Generalized Wieferich primes and monogenic polynomials

本文研究广义Wieferich素数在多项式单生成性中的应用，针对一类不依赖Lucas序列同余条件的三项式给出单生成性判据，并推广至任意$N\ge 4$的$N$-项式。

abc猜想蕴含数域中固定基的无穷多个非Wieferich位

The abc conjecture implies infinitely many non-Wieferich places for fixed bases in number fields

本文在abc猜想假设下，将非Wieferich素数增长结果推广至数域，证明Ding的结果在虚二次域整数环中除31个例外外对所有基成立。

数字段中的Wieferich素数及Ankeny-Artin-Chowla和Mordell猜想

Wieferich primes in number fields and the conjectures of Ankeny--Artin--Chowla and Mordell

本文研究数字段中Wieferich素数的性质，证明在特定条件下存在无限多个非Wieferich素数，并给出相关计数的下界。

非魏尔费尔德性质的素理想与埃罗兹猜想

Non-Wieferich property of prime ideals and a conjecture of Erdös

本文研究了素理想在特定条件下的非魏尔费尔德性质，并推广了杜普-魏尔费尔德定理，探讨了β进制展开中数字的渐近等分布及块复杂度。

关于塔库尔对卡利茨 - 维费里希素数猜想的一个反例

A counterexample to a conjecture of Thakur on Carlitz-Wieferich primes

研究塔库尔关于卡利茨 - 维费里希素数的猜想，通过构造在\(F_{19^3}\)上次数为\(5\)的不可约\(c -\)维费里希素数给出反例，还给出相关公因子封闭形式，表明次数\(5\)是最小反例次数，特定范围内素域\(F_p\)无反例。

Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771

The Fermat quotient $q_p(a):=(a^{p-1}-1)/p$, for prime $p\nmid a$, and the Wilson quotient $w_p:=((p-1)!+1)/p$ are integers. If $p\mid w_p,$ then $p$ is a Wilson prime. For odd $p,$ Lerch proved that $(\sum_{a=1}^{p-1} q_p(a) - w_p)/p$ is also an integer; we call it the Lerch quotient $\ell_p.$ If $p\mid\ell_p$ we say $p$ is a Lerch prime. A simple Bernoulli-number test for Lerch primes is proven. There are four Lerch primes 3, 103, 839, 2237 up to $3\times10^6$; we relate them to the known Wilson primes 5, 13, 563. Generalizations are suggested. Next, if $p$ is a non-Wilson prime, then $q_p(w…

第二类Mac Lane（上）同调与Wieferich素数

Mac Lane (co)homology of the second kind and Wieferich primes

本文定义第二类Mac Lane（上）同调，计算整体数环有限局部化的该不变量，揭示其与Wieferich素数无限性的等价关系，并确定Mac Lane上同调的环结构及Adams运算。

A note on balancing sequences and application to cryptography

In this paper, we prove the lower bound for the number of balancing non-Wieferich primes in arithmetic progressions. More precisely, for any given integer $r\geq2$ there are $\gg\log x$ balancing non-Wieferich primes $p\leq x$ such that $p\equiv\pm1 \pmod{r}$, under the assumption of the $abc$ conjecture for the number field $\mathbb{Q}(\sqrt{2})$. Further, we discuss some applications of balancing sequences in cryptography.

Wall's Conjecture and the ABC Conjecutre

We show that the $abc$ conjecture of Masser-Oesterlé-Szpiro for number fields implies that there are infinitely many non-Fibonacci-Wieferich primes. We also provide a new heuristic for the number of such primes beneath a certain value.

Galois groups of some iterated polynomials over cyclotomic extensions

Let $φ_p(z)=(z-1)^p+2-ζ_p$, where $ζ_p\in\bar{\mathbb{Q}}$ is a primitive $p$-th root of unity for some odd prime $p$. Building on previous work, we show that the $n$-th iterate $φ_p^n(z)$ has Galois group $[C_p]^n$, an iterated wreath product of cyclic groups, whenever $p$ is not a Wieferich prime.

A search for primes $p$ such that Euler number $E_{p-3}$ is divisible by $p$

Let $p>3$ be a prime. Euler numbers $E_{p-3}$ first appeared in H. S. Vandiver's work (1940) in connection with the first case of Fermat Last Theorem. Vandiver proved that $x^p+y^p=z^p$ has no solution for integers $x,y,z$ with $\gcd(xyz,p)=1$ if $E_{p-3}\equiv 0 (\bmod p)$. Numerous combinatorial congruences recently obtained by Z.-W. Sun and by Z.-H. Sun involve the Euler numbers $E_{p-3}$. This gives a new significance to the primes $p$ for which $E_{p-3}\equiv 0 (\bmod p)$. For the computation of residues of Euler numbers $E_{p-3}$ modulo a prime $p$, we use the congruence which runs signi…

Overpseudoprimes, and Mersenne and Fermat numbers as primover numbers

We introduce a new class of pseudoprimes-so called "overpseudoprimes to base $b$", which is a subclass of strong pseudoprimes to base $b$. Denoting via $|b|_n$ the multiplicative order of $b$ modulo $n$, we show that a composite $n$ is overpseudoprime if and only if $|b|_d$ is invariant for all divisors $d>1$ of $n$. In particular, we prove that all composite Mersenne numbers $2^{p}-1$, where $p$ is prime, are overpseudoprime to base 2 and squares of Wieferich primes are overpseudoprimes to base 2. Finally, we show that some kinds of well known numbers are overpseudoprime to a base $b$.

Powers of two weighted sum of the first p divided Bernoulli numbers modulo p

We show that, modulo some odd prime p, the powers of two weighted sum of the first p-2 divided Bernoulli numbers equals the Agoh-Giuga quotient plus twice the number of permutations on p-2 letters with an even number of ascents and distinct from the identity. We provide a combinatorial characterization of Wieferich primes, as well as of primes p for which p^2 divides the Fermat quotient q_p(2).

素数基二项式同余的结构化解

Structured Solutions of Prime-Base Binomial Congruences

研究素数基 q 下二项式同余式 C(qn, n) ≡ q^n (mod n) 的解结构，将解 n = q^t p 转化为模 p 同余和数字和不等式两个独立条件，并用于生成新解及证明平方解与 Wieferich 素数等价。

Large Sieve Inequalities for Special Characters to Prime Square Moduli

In this paper, we develop a large sieve type inequality for some special characters whose moduli are squares of primes. Our result gives non-trivial estimate in certain ranges.

Bounded gaps between primes of the special form

For each $m\geq 1$, there exist infinitely many primes $p_1<p_2<\ldots<p_{m+1}$ such that $p_{m+1}-p_1=O(m^4e^{8m})$ and $p_j+2$ has at most $\frac{16m}{\log 2}+\frac{5\log m}{\log 2}+37$ prime divisors for each $j$.

Ternary Goldbach's Problem Involving Primes of a Special Type

Let $η$ be a quadratic irrationality. The variant of a ternary problem of Goldbach involving primes such that $a<\{ηp\}<b$, where $a$ and $b$ are arbitrary numbers of the interval $(0,1)$, solved in this paper.

Hua Loo Keng's Problem Involving Primes of a Special Type

Let $η$ be a quadratic irrationality. The variant of Hua Loo Keng's problem involving primes such that $a<\{ηp^2\}<b$, where $a$ and $b$ are arbitrary real numbers of the interval $(0,1)$, solved in this paper.

Non-vanishing of special $L$-values of cusp forms on GL(2) with totally split prime power twists

We prove the non-vanishing of special $L$-values of cuspidal automorphic forms on GL(2) twisted by Hecke characters of prime power orders and totally split prime power conductors. Main ingredients of the proof are estimating the Galois averages of fast convergent series expressions of special $L$-values and considering Shintani cone decomposition.

On the distribution of αp modulo one for primes p of a special form

A classical problem in analytic number theory is to study the distribution of $αp$ modulo 1, where $α$ is irrational and $p$ runs over the set of primes. We consider the subsequence generated by the primes $p$ such that $p+2$ is an almost-prime (the existence of infinitely many such $p$ is another topical result in prime number theory) and prove that its distribution has a similar property.

On the number of ramified primes in specializations of function fields over $\mathbb{Q}$

We study the number of ramified prime numbers in finite Galois extensions of $\mathbb{Q}$ obtained by specializing a finite Galois extension of $\mathbb{Q}(T)$. Our main result is a central limit theorem for this number. We also give some Galois theoretical applications.

Diophantine approximation by special primes

We show that whenever $δ>0$, $η$ is real and constants $λ_i$ satisfy some necessary conditions, there are infinitely many prime triples $p_1,\, p_2,\, p_3$ satisfying the inequality $|λ_1p_1 + λ_2p_2 + λ_3p_3+η|<(\max p_j)^{-1/12+δ}$ and such that, for each $i\in\{1,2,3\}$, $p_i+2$ has at most $28$ prime factors.

Goldbach's Problem in Primes with Binary Expansions of a Special Form

Let $\mathbb{N}_0$ be a class of natural numbers whose binary expansions contain even numbers of ones. Goldbach's problem in numbers of class $\mathbb{N}_0$ is solved.

On the distribution of $αp^2$ modulo one with prime $p$ of a special form

Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, it is proved that for $α\in\mathbb{R}\backslash\mathbb{Q},\,β\in\mathbb{R}$ and $0<θ<10/1561$, there exist infinitely many primes $p$, such that $\|αp^2+β\|<p^{-θ}$ and $p+2=\mathcal{P}_4$, which constitutes an improvement upon the previous result.

A quaternary diophantine inequality by prime numbers of a special type

Let $1<c<832/825$. For large real numbers $N>0$ and a small constant $\vartheta>0$, the inequality \begin{equation*} |p_1^c+p_2^c+p_3^c+p_4^c-N|<\vartheta \end{equation*} has a solution in prime numbers $p_1,\,p_2,\,p_3,\,p_4$ such that, for each $i\in\{1,2,3,4\}$, $p_i+2$ has at most $32$ prime factors.

On an equation involving fractional powers with prime numbers of a special type

We consider the equation $[p_{1}^{c}] + [p_{2}^{c}] + [p_{3}^{c}] = N$, where $N$ is a sufficiently large integer, and prove that if $1 < c < \frac{17}{16}$, then it has a solution in prime numbers $p_{1}$, $p_{2}$, $p_{3}$ such that each of the numbers $p_{1} + 2$, $p_{2} + 2$, $p_{3} + 2$ has at most $\left [ \frac{95}{17 - 16c} \right ]$ prime factors, counted with the multiplicity.

Bounded gaps between primes in special sequences

We use Maynard's methods to show that there are bounded gaps between primes in the sequence $\{\lfloor nα\rfloor\}$, where $α$ is an irrational number of finite type. In addition, given a superlinear function $f$ satisfying some properties described by Leitmann, we show that for all $m$ there are infinitely many bounded intervals containing $m$ primes and at least one integer of the form $\lfloor f(q)\rfloor$ with $q$ a positive integer.

关于整数表示为无平方因子数与特殊类型素数之和

On the representation of integer as sum of a square-free number and a prime of special type

本文证明存在无穷多个整数可表示为一个无平方因子整数与一个满足特定丢番图不等式条件的素数之和，其中系数α为无理数。

Strings of special primes in arithmetic progressions

The Green-Tao Theorem, one of the most celebrated theorems in modern number theory, states that there exist arbitrarily long arithmetic progressions of prime numbers. In a related but different direction, a recent theorem of Shiu proves that there exist arbitrarily long strings of consecutive primes that lie in any arithmetic progression that contains infinitely many primes. Using the techniques of Shiu and Maier, this paper generalizes Shiu's Theorem to certain subsets of the primes such as primes of the form $\lfloor πn\rfloor$ and some of arithmetic density zero such as primes of the form $…

Special Sets of Primes in Function Fields

When investigating the distribution of the Euler totient function, one encounters sets of primes P where if p is in P then r is in P for all r|(p-1). While it is easy to construct finite sets of such primes, the only infinite set known is the set of all primes. We translate this problem into the function field setting and construct an infinite such set in F_p[x] whenever p is equivalent to 2 or 5 modulo 9.

On a diophantine inequality with prime numbers of a special type

We consider the Diophantine inequality \[ \left| p_1^{c} + p_2^{c} + p_3^c- N \right| < (\log N)^{-E} , \] where $1 < c < \frac{15}{14}$, $N$ is a sufficiently large real number and $E>0$ is an arbitrarily large constant. We prove that the above inequality has a solution in primes $p_1$, $p_2$, $p_3$ such that each of the numbers $p_1 + 2, p_2 + 2, p_3 + 2$ has at most $\left[ \frac{369}{180 - 168 c} \right]$ prime factors, counted with the multiplicity.

The ternary Goldbach problem with a prime with a missing digit and primes of special types

Let $$γ^*:=\frac{8}{9}+\frac{2}{3}\:\frac{\log(10/9)}{\log 10}\:(\approx 0.919\ldots)\:,\ γ^*<\frac{1}{c_0}\leq 1\:.$$ Let $γ^*<γ_0\leq 1$, $c_0=1/γ_0$ be fixed. Let also $a_0\in\{0,1,\ldots, 9\}$. In [23] we proved on assumption of the Generalized Riemann Hypothesis (GRH), that each sufficiently large odd integer $N_0$ can be represented in the form $$N_0=p_1+p_2+p_3\:,$$ where for $i=2, 3$ the primes $p_i$ are Piatetski-Shapiro primes - primes of the form $p_i=[n_i^{c_0}]$, $n_i\in\mathbb{N}$ - whereas the decimal expansion of $p_1$ does not contain the digit $a_0$. In this paper we replace …

关于特殊序列中的素数及其在卡迈克尔数中的应用

On primes in special sequences with applications to Carmichael numbers

本文利用等差数列中与Λ(n)相关的指数和，改进了非齐次Beatty序列和Piatetski-Shapiro序列上素数分布的结果，并将其应用于改进卡迈克尔数相关结论。

Constructing Certain Special Analytic Galois Extensions

For every prime $p\geq 5$ for which a certain condition on the class group $\text{Cl}(\mathbb{Q}(μ_p))$ is satisfied, we construct a $p$-adic analytic Galois extension of the infinite cyclotomic extension $\mathbb{Q}(μ_{p^{\infty}})$ with some special ramification properties. In greater detail, this extension is unramified at primes above $p$ and tamely ramified above finitely many rational primes and is isomorphic to a finite index subgroup of $\text{SL}_2(\mathbb{Z}_p)$ which contains the principal congruence subgroup. For the primes $107,139,271$ and $379$ such extensions were first constru…

Constellations of primes generated by twin primes

We study a special set of constellations of primes generated by twin primes.

关于特殊类型素数模一的 $\sqrt{p}$ 分布的一个注记

A remark on the distribution of $\sqrt{p}$ modulo one involving primes of special type II

研究特殊类型素数模一的平方根分布问题，证明不等式有无穷多素数解，改进了已有结果。

Exceptional Prime Number Twins, Triplets and Multiplets

A classification of twin primes implies special twin primes. When applied to triplets, it yields exceptional prime number triplets. These generalize yielding exceptional prime number multiplets.

Shimura曲线上CM点在分歧素数模下的等分布

Equidistribution of CM points on a Shimura Curve modulo a ramified prime

本文利用Ratner定理研究了完全实数域上Shimura曲线特殊纤维中CM点的约化，证明了CM点Galois轨道在分歧素数模下的等分布命题。

关于一类涉及特殊类型素数的二元丢番图不等式

On a binary Diophantine inequality involving primes of a special type

本文研究了涉及特殊类型素数的丢番图不等式可解性问题，证明了二元与四元情形下，满足对应素因子个数上界条件的素数解的存在性，给出了素因子数的具体上界公式。

特殊集合中任意长的连续素数串

Arbitrarily long strings of consecutive primes in special sets

本文研究一类由广义多项式小数部分落入非空开集所定义的自然数集合，证明其中存在任意长的连续素数串，揭示了此类特殊集合在素数分布上的强正则性。

Special Values of Anticyclotomic L-functions Modulo λ

The purpose of this article is to generalize some results of Vatsal on studying the special values of Rankin-Selberg L-functions in an anticyclotomic $\mathbb{Z}_{p}$-extension. Let $g$ be a cuspidal Hilbert modular form of parallel weight (2,...,2) and level $\mathcal{N}$ over a totally real field F, and let K/F be a totally imaginary quadratic extension of relative discriminant $\mathcal{D}$. We study the $l$-adic valuation of the special values $L(g,χ,\frac{1}{2})$ as χvaries over the ring class characters of K of $\mathcal{P}$-power conductor, for some fixed prime ideal $\mathcal{P}$. We p…

Cyclicity of Elliptic Curves Modulo Primes in arithmetic progressions

We consider the reduction of an elliptic curve defined over the rational numbers modulo primes in a given arithmetic progression and investigate how often the subgroup of rational points of this reduced curve is cyclic as a special case of Serre's Cyclicity Conjecture.

An alternative proof of the Dirichlet prime number theorem

Dirichlet's theorem on arithmetic progressions called as Dirichlet prime number theorem is a classical result in number theory. Atle Selberg\cite{Selberg} gave an elementary proof of this theorem. In this article we give an alternative proof of it based on a previous result of us. Also we get an estimation of the prime counting function in the special cases.

On non-congruent numbers with 1 modulo 4 prime factors

In this paper, we use the 2-decent method to find a series of odd non-congruent numbers $\equiv1\pmod 8$ whose prime factors are $\equiv1\pmod4$ such that the congruent elliptic curves have second lowest Selmer groups, which includes Li and Tian's result (Li and Tian, 2000) as special cases.

$\u00a0\mathrm{GL}(2)$\u00a0上尖点形式的特殊\u00a0$L$\u00a0值在完全分裂素数幂扭下的非零性：无需类数假设

Nonvanishing of special $L$-values of cusp forms on $\mathrm{GL}(2)$ with totally split prime power twists: without the class number hypothesis

本文证明在完全分裂素数幂扭下，GL(2) 上尖点形式的特殊 L 值在中心临界点非零，无需类数假设，改进了 Kwon 和 Sun 的结果。