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Results tagged with nt.number-theory
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user 3684
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
1
vote
$\zeta(s+1)/\zeta(s)$
If I understand correctly, what Scott wants is a citation for Franel's paper on (Farey series and) the Riemann Hypothesis. That would be Les suites de Farey et le problème des nombres premiers, Göttin …
- 38.5k
5
votes
Representations as a sum of cubes following Jacobi
I'm sure the sums you write down are used, in conjunction with the circle method, to find asymptotic expressions for the number of representations as sums of cubes, but if it were possible to do for c …
- 38.5k
3
votes
Minimum number of twin primes < N
Let $f(N)$ be the number of known twin primes up to $N$. Then I'm afraid the only known lower bound for the number of twin primes up to $N$ is $f(N)$. If no one has ever looked to see whether there ar …
- 38.5k
2
votes
Need to require a assertion for two subsets of natural numbers
T C Brown and Peter Jau-shyong Shiue, A remark related to the Frobenius problem, Fibonacci Quart. 31 (1993) 32-36, MR 93k:11018 gives you what you need.
"For given $a,b$ with $(a, b) = 1$, let $NR(a …
- 38.5k
10
votes
sums of fractional parts of linear functions of n
Let $C_m(\alpha)=\sum_{k=1}^m((k\alpha))$ where $((x))$ is $x-[x]-1/2$ if $x$ is not an integer, 0 if $x$ is an integer (so this agrees with your definition away from points where $k\alpha$ is an inte …
- 38.5k
14
votes
Accepted
Range of $2^n$ mod $n$
The smallest such $n$ is $n=4700063497$. A few others are known. J. Crump found $n=8365386194032363$ in 2000. Max Alekseyev found $n=3468371109448915$. Joe Crump found $n=10991007971508067$. Some info …
- 38.5k
1
vote
Questions about normal numbers
I think the answer to the 2nd question is yes (provided I understand the question). Take any normal sequence in any base $b>1$. Change the 1st symbol (if necessary) so the sequence begins with 1. Chan …
- 38.5k
0
votes
Spacing of zeros of zeta function on the critical line
I can't figure out what $\Lambda(k)$ is. The author defines $\Lambda$ by (1.4) on page 3, and it's an absolute constant. Then $\Lambda$ as a function shows up for the first time at (1.18) on page 6, w …
- 38.5k
2
votes
Distribution mod 1 of Factorial Multiples of Real Numbers.
I'll have to look this up in Kuipers and Niederreiter to make sure I have it right, but I think that for any increasing sequence $a_1,a_2,\dots$ of integers the sequence $a_1x,a_2x,\dots$ is uniformly …
- 38.5k
3
votes
S(3) field extensions
If $f$ is a cubic, irreducible over the rationals, then its splitting field has Galois group cyclic of order three if the discriminant is a square, symmetric on three letters otherwise. I don't know e …
- 38.5k
7
votes
Why are the only numbers $m$ for which $n^{m+1}\equiv n \bmod m$ also the only numbers such ...
We have, for both congruences, if $p$ is a prime dividing $m$, then $p-1$ also divides $m$, and $p^2$ doesn't divide $m$; conversely, if $m$ satisfies these properties, then it works in both congruenc …
- 38.5k
5
votes
Accepted
Distribution of integers with number of prime factors lying in a given arithmetic progression
What you call $\lambda(n)$ is often denoted $\Omega(n)$ in the literature.
Hubert Delange, Sur la distribution des valeurs de certaines fonctions arithmétiques, Colloque sur la Théorie des Nombres, …
- 38.5k
6
votes
Accepted
Covering systems
A covering system (or, simply, a cover) is a finite collection of congruences $x\equiv a_i\pmod{m_i}$ with distinct moduli, each modulus exceeding 1, such that every integer satisfies at least one of …
- 38.5k
5
votes
factorazy $k$-tuples
The $k=3$ case is solved in Erdos & Pomerance, On the largest prime factors of $n$ and $n+1$, Aequationes Mathematicae 17 (1978) 311-321, available here. The solution is short enough to give here in i …
- 38.5k
2
votes
Accepted
How long can a primal egyptian fraction be, that optimally approaches unity?
This may be related to Giuga numbers: composite numbers $n$ such that $p$ divides $(n/p)-1$ for every prime divisor $p$ of $n$. A 10-factor Giuga number is given in the comments on that page: $$\eqali …
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