Generalizations of Carmichael numbers I

May 4, 2013 - Abstract. A composite positive integer n is said to be a weak Carmichael number if ... that ϕ(n) | (n−1) [48], where ϕ(n) is the Eul...

0 downloads 16 Views 448KB Size

Recommend Documents

Aug 20, 2015 - A . It is shown that the set of decimal palindromes is an additive basis ... Every number n P N has a unique decimal representation of the form.

whenever d P B X r1,xBys, |µpdq| “ 1, P`pdq,y ď xB{ log2 x and pd, ℓpxBqq “ 1. D .... that of [1, Theorem 4.1] verbatim, so let us only set up the proof here. Let.

known results; see, e.g., Iwaniec and Kowalski [8, Theorem 12.6]). Further advances ... applications to L-functions and to the distribution of primes in progressions modulo pγ. Iwaniec [7] has extended those results to moduli q with a small core q7.

Feb 20, 2016 - be the number of integers n P rM `1,M `Ns for which Q´aupnq ..... Squaring out the right side and estimating the contribution from diagonal.

function satisfies an estimate of the form. πP pxq “ δ πpxq ` O`plog log xqεpxq˘, we show that P is an exact asymptotic additive basis for N, i.e., for some integer h “ hpPq ą 0 the sumset hP contains all but finitely many natural numbers.

Jun 10, 2015 - I. For any prime number n, Fermat's little theorem asserts that an ” a pnq pa P Zq. (1.1). Around 1910, Carmichael initiated the study of composite numbers n with the property (1.1); these are now known as Carmichael numbers. The exi

May 28, 2017 - where ⌊·⌋ denotes the floor function: ⌊t⌋ is the greatest integer ⩽ t for any t .... The notation 〈〈t〉〉 is used to represent the distance from the real ...

Feb 1, 2018 - Equation (12)] (and also [13] for the best known bound under the Riemann. Hypothesis). According to ... stronger than (1.5) and gives a better saving with a higher power of log x in the exponent;. ‚ valid for ... which we believe can

Sep 26, 2017 - use of the Euler product for D(s). Instead, a combinatorial identity is employed to verify that the last condition of Theorem 1.1 is met. Theorem 1.2. Let D(s) = ∑n∈N F(n)n−s, where F is completely multiplicative. Then the Dirich

Aug 18, 2011 - Classically, a composite integer n > 2 is called a pseudoprime to the base b if ..... ǫ(E(Z/piZ)) = {Api if the sequence (5.7) does not split,. Api−1 if the sequence (5.7) does split. .... The non-square-free numbers in this list ar

Sep 1, 2016 - arXiv:1609.00231v1 [math.NT] 1 Sep ... isomorphic to an order in an imaginary quadratic field Q(√−d) with class number. 1. We denote this ... Definition 2.2. Let n be composite. If, for a given elliptic curve E, n satisfies the Gord

Sep 5, 2013 - The symbol log will denote the base 2 logarithm, while ln denotes the natural ... thors credit their inspiration to [18], who first used the Erd˝os ...

Dec 24, 2012 - Fermat's Little Theorem asserts that if m is prime then ... After the proof of the infinitude of Carmichael numbers in [AGP], focus began to.

and |1 > qubits with coefficients being complex numbers α and β, i.e. with states |ψ ... ber of bases 1 < a < k coprime to k and which sat- isfy eq. (1), i.e. for which k ...

Sep 14, 2018 - Using Theorem 2.6, it is easy to get a necessary and sufficient con- dition for an ..... Finally, the desired result follows from (4.4), (4.7) and (4.11). D. Corollary ... and some partial answers to the second and third questions whos

Feb 26, 2017 - Section 3.2 discusses this algorithm, which uses a Monte Carlo simulation to check if a composite number is .... bounded above by 50%, the observations pertaining to the rate of Fermat witnesses for. Carmichael numbers ..... if gcd(sam

Jan 3, 2017 - Abstract. This paper studies the generalizations of the Stirling numbers of both kinds and the Lah numbers in association with the normal order problem in the Weyl algebra. W = 〈x, D|Dx − xD = 1〉. Any word ω ∈ W with m x's and

Aug 22, 2015 - Abstract. In this paper, we prove that there are infinitely many n for which rad(ϕ(n))|n − 1 but n is not a Carmichael number. Additionally, we ...

Dec 17, 2012 - Hassan Jolany and M.R.Darafsheh, R.Eizadi Alikelaye ..... [6] Y.Hamahata,H.Masubuch:Special Multi-Poly- Bernoulli numbers ,Journal of ...

Jun 24, 2011 - basic identities satisfied by the harmonic numbers. For example,. Email address: [email protected] (István Mez˝o). URL: (István Mez˝o). 1 Present address:

Dec 2, 2015 - set of composite numbers which in effect fool the Fermat test, causing it to report that they are prime. These numbers are ... Instead of checking the whole sequence (1), only check the last z + 1 numbers. ... simply guessed that there

Jun 22, 2017 - primes without large prime factors. Abstract. We prove some new lower bounds for the counting function NC(x) of the set of Novák-Carmichael ...

May 29, 2014 - λ(e)(pa)xa =1+ x + x2 + 2x3 + 2x4 + 4x5 + 2x6 + 6x7 + O(x8). = 1 + O(x8). (1 − x)(1 − x3)(1 ..... National University. E-mail address: [email protected]

Jul 27, 2017 - number and t ≥ 3. It seems that the sequence (mn)n≥0 is always periodic for any p and t. Computer calculation has provided the first values: ...... Discrete. Math., 205(1-3):119–143, 1999. [12] Donald E. Knuth. The art of compute