Alex Samuel Bamunoba, Jonas Bergström
Let $q$ be a power of a prime number $p$, $\mathbb F_q$ be a finite field with $q$ elements and $\mathcal{G}$ be a subgroup of $(\mathbb F_q,+)$ of order $p$. We give an existence criterion and an algorithm for computing maximally $\mathcal{G}$-fixed c-Wieferich primes in $\mathbb F_q[T]$. Using the criterion, we study how c-Wieferich primes behave in $\mathbb F_q[T]$ extensions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We prove that there are no Wieferich's primes $q=2p+1$ where $p \equiv 3 \pmod{4}$ is a prime number
In this paper, we introduce a Carlitz module analogue of Mersenne primes, and prove Carlitz module analogues of several classical results concerning Mersenne primes. In contrast to the classical case, we can show that there are infinitely many composite Mersenne numbers. We also study the acquaintances of Mersenne primes including Wieferich and non-Wieferich primes in the Carlitz module context that were first introduced by Dinesh Thakur.
Wieferich and Mersenne primes for function fields
本文研究了函数域中魏弗里希素数与梅森数的性质及其与费马方程的关系。
Regular primes, non-Wieferich primes, and finite multiple zeta values of level $N$
本文引入一般级的有限多重zeta值，研究其非零性与正则素数及非Wieferich素数的联系，并提出若干更易处理的相关问题以替代证明素数无穷性的困难目标。
In this paper, we define $X$-base Fibonacci-Wieferich prime which is a generalized Wieferich prime where $X$ is a finite set of algebraic numbers. We are going to show that there are infinitely many non-$X$-base Fibonacci-Wieferich primes assuming the $abc$-conjecture of Masser-Oesterlé-Szpiro for number fields. We also provide a new conjecture concerning the rank of free part of abelian group generated by all elements in $X$, and we will use the arithmetic point of view and geometric point of view to give heuristic.
We prove the lower bound for the number of Lucas non-Wieferich primes in arithmetic progressions. More precisely, for any given integer $k\geq 2$ there are $\gg \log x$ Lucas non-Wieferich primes $p\leq x$ such that $p\equiv\pm1\pmod{k}$, assuming the $abc$ conjecture for number fields. Further, we discuss some applications of Lucas sequences in Cryptography.
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…
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.
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}。
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 …
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.
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 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。
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…
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。
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。
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。
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。
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。
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。
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…
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。
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.
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.
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.
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…
Let $S$ be a set of primes. We call an $m$-tuple $(a_1,\ldots,a_m)$ of distinct, positive integers $S$-Diophantine, if for all $i\neq j$ the integers $s_{i,j}:=a_ia_j+1$ have only prime divisors coming from the set $S$, i.e. if all $s_{i,j}$ are $S$-units. In this paper, we show that no $S$-Diophantine quadruple (i.e.~$m=4$) exists if $S=\{3,q\}$. Furthermore we show that for all pairs of primes $(p,q)$ with $p<q$ and $p\equiv 3\mod 4$ no $\{p,q\}$-Diophantine quadruples exist, provided that $(p,q)$ is not a Wieferich prime pair.
Let $p \ge 5$ be a prime and let $\mathbb{Q}_{n,p}$ denote the $n$-th layer of the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$. We show that $\mathbb{Q}_{n,p}$ has no exceptional units. We use this to prove the effective asymptotic Fermat's Last Theorem over $\mathbb{Q}_{n,p}$ for all $n \ge 1$ and all primes $p \ge 5$ that are non-Wieferich, i.e. $2^{p-1} \not \equiv 1 \pmod{p^2}$. The effectivity in our result builds on recent work of Thorne proving modularity of elliptic curves over $\mathbb{Q}_{n,p}$.
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 $b$.
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).
Some new infinite families of non-$p$-rational real quadratic fields
受量子信息SIC-POVMs问题研究启发，提出构造同时非p_j-有理、p_j上方非分歧的实二次域无限族的简单方法，可得到p幂循环分量任意大的相关域。
On the Non p-Rationality and Iwasawa Invariants of Certain Real Quadratic Fields
本文研究实二次域$\mathbb{Q}(\sqrt{m^2p^{2r}+1})$的非$p$-有理性，证明在$m$小于某常数时基本单位满足特定同余式从而得到无限族非$p$-有理域，并利用Fukuda-Komatsu准则给出岩泽不变量为零的充分条件。
The use of the invariant's properties in the primality test and prime search
本文探讨了不变量性质在素性测试与素数搜索中的应用，提出了基于该性质的新算法，分析了其优缺点，并指出其能提供复合数因式分解的额外信息。
The results of the computer hunt for the primes of the form $q = m^2+1$ up to $10^{20}$ are reported. The number of sign changes of the difference $π_q(x) - \frac{C_q}{2}\int_2^x{du \over \sqrt{u}\log(u)}$ and the error term for this difference is investigated. The analogs of the Brun's constant and the Skewes number are calculated. An analog of the B conjecture of Hardy--Littlewood is formulated. It is argued that there is no Chebyshev bias for primes of the form $q=m^2+1$. All encountered integrals we were able to express by the logarithmic integral.
A prime $p$ is a Wolstenholme prime if $\binom{2p}{p}\equiv2$ mod $p^4$, or, equivalently, if $p$ divides the numerator of the Bernoulli number $B_{p-3}$; a Vandiver prime $p$ is one that divides the Euler number $E_{p-3}$. Only two Wolstenholme primes and eight Vandiver primes are known. We increase the search range in the first case by a factor of $10$, and show that no additional Wolstenholme primes exist up to $10^{11}$, and in the second case by a factor of $20$, proving that no additional Vandiver primes occur up to this same bound. To facilitate this, we develop a number of new congruen…
A Wilson prime is a prime p such that (p-1)! = -1 mod p^2. We report on a search for Wilson primes up to 2 * 10^13, and describe several new algorithms that were used in the search. In particular we give the first known algorithm that computes (p-1)! mod p^2 in average polynomial time per prime.
Kurepa's conjecture states that there is no odd prime $p$ that divides $!p=0!+1!+\cdots+(p-1)!$. We search for a counterexample to this conjecture for all $p<2^{34}$. We introduce new optimization techniques and perform the computation using graphics processing units. Additionally, we consider the generalized Kurepa's left factorial given by $!^{k}n=(0!)^k +(1!)^k +\cdots+((n-1)!)^{k}$, and show that for all integers $1<k<100$ there exists an odd prime $p$ such that $p\mid !^k p$.
Motivated by Euler's observation that the polynomial $x^{2} + x + 41$ takes on prime values for $0 \leq x \leq 39$, we search for large values of $x$ for which $N = x^{2} + x + 41$ is prime. To apply classical primality proving results based on the factorization of $N-1$, we choose $x$ to have the form $g(y)$, chosen so that $g(y)^{2} + g(y) + 40$ is reducible. Our main result is an explicit, 60,000 digit prime number of the form $x^{2} + x + 41$.
$13$ unknowns over quadratic integer rings and Lucas congruences
该研究针对二次数域，证明了整数元组的3个未知元丢番图定义，得到13个未知元的递归可枚举整数关系表示，进而得出存在次数界D₀使得无法判定对应多项式方程在二次整数环上是否有解的结论。
We describe a search for primitive trinomials of high degree and its interaction with the Great Internet Mersenne prime search (GIMPS). The search is complete for trinomials whose degree is the exponent of a Mersenne prime, for all 47 currently known Mersenne primes.
We present new generalizations of Olson's theorem and of a consequence of Alon's Combinatorial Nullstellensatz. These enable us to extend some of their combinatorial applications with conditions modulo primes to conditions modulo prime powers. We analyze computational search problems corresponding to these kinds of combinatorial questions and we prove that the problem of finding degree-constrained subgraphs modulo $2^d$ such as $2^d$-divisible subgraphs and the search problem corresponding to the Combinatorial Nullstellensatz over $\mathbb{F}_2$ belong to the complexity class Polynomial Parity…
From Worst to Average Case to Incremental Search Bounds of the Strong Lucas Test
本文研究强Lucas测试的错误界，证明通过t轮连续测试及额外标准测试的整数为素数的概率大于1-(4/15)^t，并首次给出基于强Lucas测试的增量搜索算法的错误界。
In this paper, we discuss an improvement of an algorithm to search for primes $p$ and coset-partitions of Z/pZ* that yield Ramsey algebras over Z/pZ. We also prove an upper bound on the modulus p in terms of the number of cosets. We have, as a corollary, that there is no prime $p$ for which there exists a partition of Z/pZ* into 13 cosets that yields a 13-color Ramsey algebra. Thus A263308(13) = 0.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。