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On Generalized Carmichael Numbers
, 2000
"... . For arbitrary integers k 2 Z we investigate the set C k of the generalized Carmichael numbers, i.e. the natural numbers n ? maxf1; 1 \Gamma kg such that the equation a n+k j a mod n holds for all a 2 N. We give a characterization of these generalized Carmichael numbers and discuss several spe ..."
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. For arbitrary integers k 2 Z we investigate the set C k of the generalized Carmichael numbers, i.e. the natural numbers n ? maxf1; 1 \Gamma kg such that the equation a n+k j a mod n holds for all a 2 N. We give a characterization of these generalized Carmichael numbers and discuss several special cases. In particular, we prove that C 1 is finite and that C k is infinite, whenever 1 \Gamma k ? 1 is squarefree. We also discuss generalized Carmichael numbers which have one or two prime factors. Finally, we consider the Jeans numbers, i.e. the set of odd numbers n which satisfy the equation a n j a mod n only for a = 2, and the corresponding generalizations. We give a stochastic argument which supports the conjecture that infinitely many Jeans numbers exist which are squares. 1 Introduction: Historical Background On October 18th, 1640, Pierre de Fermat wrote in a letter to Bernard Frenicle de Bessy that if p is a prime number, then p divides a p\Gamma1 \Gamma 1 for all in...
AN UPPER ESTIMATE FOR THE OVERPSEUDOPRIME COUNTING FUNCTION
, 807
"... Abstract. We show that the number of overpseudoprimes to base 2 not exceeding x is o(x ε), where ε> 0 is arbitrary small for sufficiently large x. 1. ..."
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Abstract. We show that the number of overpseudoprimes to base 2 not exceeding x is o(x ε), where ε> 0 is arbitrary small for sufficiently large x. 1.
Ramanujan and Labos primes, their generalizations, and classifications of primes
 J. Integer Seq
"... We study the parallel properties of the Ramanujan primes and a symmetric counterpart, the Labos primes. Further, we study all primes with these properties (generalized Ramanujan and Labos primes) and construct two kinds of sieves for them. Finally, we give a further natural generalization of these c ..."
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We study the parallel properties of the Ramanujan primes and a symmetric counterpart, the Labos primes. Further, we study all primes with these properties (generalized Ramanujan and Labos primes) and construct two kinds of sieves for them. Finally, we give a further natural generalization of these constructions and pose some conjectures and open problems. 1
PROCESS OF ”PRIMOVERIZATION ” OF NUMBERS OF THE FORM a n − 1
, 807
"... Abstract. We call an integer N> 1 primover to base a if it either prime or overpseudoprime to base a. We prove, in particular, that every Fermat number is primover to base 2. We also indicate a simple process of receiving of primover divisors of numbers of the form a n − 1. 1. ..."
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Abstract. We call an integer N> 1 primover to base a if it either prime or overpseudoprime to base a. We prove, in particular, that every Fermat number is primover to base 2. We also indicate a simple process of receiving of primover divisors of numbers of the form a n − 1. 1.
ON SMALL INTERVALS CONTAINING PRIMES
, 908
"... Abstract. Let p be an odd prime, such that pn < p/2 < pn+1, where pn is the nth prime. We study the following question: with what probability does there exist a prime in the interval (p, 2pn+1)? After the strong definition of the probability with help of the Ramanujan primes ([11], [12])and the int ..."
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Abstract. Let p be an odd prime, such that pn < p/2 < pn+1, where pn is the nth prime. We study the following question: with what probability does there exist a prime in the interval (p, 2pn+1)? After the strong definition of the probability with help of the Ramanujan primes ([11], [12])and the introducing pseudoRamanujan primes, we show, that if such probability P exists, then P ≥ 0.5. We also study a symmetrical case of the left intervals, which connected with sequence A080359 in [10]. 1.
ON CRITICAL SMALL INTERVALS CONTAINING PRIMES
, 908
"... Abstract. Let p be an odd prime, such that pn < p/2 < pn+1, where pn is the nth prime. We study the following question: with what probability P there exists a prime in the interval (p, 2pn+1)? We show, that for p tends to the infinity, P ≥ 1 2 (1−ε) and conjecture that P ≤ ..."
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Abstract. Let p be an odd prime, such that pn < p/2 < pn+1, where pn is the nth prime. We study the following question: with what probability P there exists a prime in the interval (p, 2pn+1)? We show, that for p tends to the infinity, P ≥ 1 2 (1−ε) and conjecture that P ≤
OVERPSEUDOPRIMES, MERSENNE NUMBERS AND WIEFERICH PRIMES
, 806
"... Abstract. We introduce a new class of pseudoprimesso called ”overpseudoprimes” which is a special subclass of superPoulet pseudoprimes. Denoting via h(n) the multiplicative order of 2 modulo n,we show that odd number n is overpseudoprime if and only if the value of h(n) is invariant of all divisor ..."
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Abstract. We introduce a new class of pseudoprimesso called ”overpseudoprimes” which is a special subclass of superPoulet pseudoprimes. Denoting via h(n) the multiplicative order of 2 modulo n,we show that odd number n is overpseudoprime if and only if the 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. 1.
ON AN INVARIANT OF DIVISORS OF MERSENNE NUMBER
, 806
"... Abstract. We prove that every divisor d> 1 of a Mersenne number 2 p − 1, where p is a prime, is a solution of the equation ordx2 = p, and introduce a special subclass of superPoulet pseudoprimes containing all Mersenne numbers. ..."
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Abstract. We prove that every divisor d> 1 of a Mersenne number 2 p − 1, where p is a prime, is a solution of the equation ordx2 = p, and introduce a special subclass of superPoulet pseudoprimes containing all Mersenne numbers.
Stream Ciphers and Linear Complexity
"... I have quite a fruitful and happy experience. My life would not be the same without my supervisor and friends. Many thanks to my supervisor, Professor Harald Niederreiter. Being a leading researcher, he is more like an intelligent gentleman for me. I am grateful for Professor Niederreiter’s directio ..."
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I have quite a fruitful and happy experience. My life would not be the same without my supervisor and friends. Many thanks to my supervisor, Professor Harald Niederreiter. Being a leading researcher, he is more like an intelligent gentleman for me. I am grateful for Professor Niederreiter’s directions on my research and study, encouragement on my work, great patience and kindly help for my thesis and graduate application. I also would like to express my sincere thanks to Professor Alan Jon Berrick, Professor
Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.3 On Intervals (kn,(k +1)n) Containing a Prime
"... We study values of k for which the interval (kn,(k + 1)n) contains a prime for every n> 1. We prove that the list of such integers k includes 1,2,3,5,9,14 and no others, at least for k ≤ 100,000,000. Moreover, for every known k in this list, we give a good upper bound for the smallest Nk(m), such th ..."
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We study values of k for which the interval (kn,(k + 1)n) contains a prime for every n> 1. We prove that the list of such integers k includes 1,2,3,5,9,14 and no others, at least for k ≤ 100,000,000. Moreover, for every known k in this list, we give a good upper bound for the smallest Nk(m), such that if n ≥ Nk(m), then the interval (kn,(k +1)n) contains at least m primes. 1 Introduction and main results In 1850, Chebyshev proved the famous Bertrand postulate (1845) that every interval [n,2n] contains a prime (for a very elegant version of his proof, see Redmond [10, Theorem 9.2]). 1 Other nice proofs were given by Ramanujan in 1919 [8] and Erdős in 1932 (reproduced in Erdős and Surányi [4, p. 171–173]). In 2006, El Bachraoui [1] proved that every interval