Arithmetic progressions of Carmichael numbers in a reduced residue class
نویسندگان
چکیده
Fix coprime natural numbers a,q. Assuming the Prime k-tuple Conjecture, we show that there exist arbitrarily long arithmetic progressions of Carmichael numbers, each which lies in reduced residue class a mod q and is product three distinct prime numbers.
منابع مشابه
Carmichael Numbers in Arithmetic Progressions
We prove that when (a, m) = 1 and a is a quadratic residue mod m, there are infinitely many Carmichael numbers in the arithmetic progression a mod m. Indeed the number of them up to x is at least x1/5 when x is large enough (depending on m). 2010 Mathematics subject classification: primary 11N25; secondary 11A51.
متن کاملOn Carmichael numbers in arithmetic progressions
Assuming a weak version of a conjecture of Heath-Brown on the least prime in a residue class, we show that for any coprime integers a and m > 1, there are infinitely many Carmichael numbers in the arithmetic progression a mod m.
متن کاملPalindromic Numbers in Arithmetic Progressions
Integers have many interesting properties. In this paper it will be shown that, for an arbitrary nonconstant arithmetic progression {an}TM=l of positive integers (denoted by N), either {an}TM=l contains infinitely many palindromic numbers or else 10|aw for every n GN. (This result is a generalization of the theorem concerning the existence of palindromic multiples, cf. [2].) More generally, for...
متن کاملPrime Numbers in Certain Arithmetic Progressions
We discuss to what extent Euclid’s elementary proof of the infinitude of primes can be modified so as to show infinitude of primes in arithmetic progressions (Dirichlet’s theorem). Murty had shown earlier that such proofs can exist if and only if the residue class (mod k ) has order 1 or 2. After reviewing this work, we consider generalizations of this question to algebraic number fields.
متن کاملSub-Ramsey Numbers for Arithmetic Progressions
Let the integers 1, . . . , n be assigned colors. Szemerédi’s theorem implies that if there is a dense color class then there is an arithmetic progression of length three in that color. We study the conditions on the color classes forcing totally multicolored arithmetic progressions of length 3. Let f(n) be the smallest integer k such that there is a coloring of {1, . . . , n} without totally m...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2021
ISSN: ['0022-314X', '1096-1658']
DOI: https://doi.org/10.1016/j.jnt.2021.03.003