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Results tagged with nt.number-theory
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user 9924
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
9
votes
Divisibility condition implies $a_1=\dotsb=a_k$?
As a partial solution, $a_1=a_2=1$, $a_3=2$ is a "counterexample for $n$ odd" in the sense that in this case there exists $N$ co-prime with $n$ such that $a_1^N+a_2^N+a_3^N$ is a multiple of $a_1^n+a …
- 22.6k
17
votes
Accepted
Existence of range of numbers containing a coprime to given n
This is tightly related to the Jacobsthal function defined to be the smallest integer $j(n)$ such that any segment of $j(n)$ consecutive integers contains an integer co-prime with $n$. Iwaniec ("On th …
- 22.6k
3
votes
Accepted
Existence of certain coprime numbers
What you seem to request is that the matrix $M=\begin{pmatrix}m_1 & -n_1 \\
m_2 & -n_2\end{pmatrix}$ were non-degenerate, and the vector $d:=M\binom ab$
had its coordinates of size polynomial in $m$. …
- 22.6k
11
votes
The Second Moment of a Sum of Floor Functions
Sums of this sort are well known to the experts - but since none of them have answered so far, let me try. Denoting the fractional part of a real $x$ by $\{x\}$, you can write your sum as
$$ S = n^2 …
- 22.6k
14
votes
Accepted
Set of integers having finite intersection with the image of any polynomial of degree $\geq 2$
There are countably many polynomials with integer coefficients of degree at least $2$; write them all in a sequence as $P_1,P_2,P_3,...\ $ For every $k\ge 1$, the set $A_k$ of all positive integers wh …
- 22.6k
5
votes
Uniformly small sums of roots of unity
This is a reworked version of my original answer, which now solves the problem for infinitely many pairs $(n,k)$ ( but certainly not all pairs).
Let $G:={\mathbb Z}/n{\mathbb Z}$. You want to show th …
- 22.6k
16
votes
How $a+b$ can grow when $a!b! \mid n!$
One has
$$ a+b < n+O(\log n), $$
which is best possible in view of $1!n!|n!$. For the proof, assuming that $a!b!|n!$ with $a+b>n$, consider the exponent of $2$ in the prime decomposition of the quo …
- 22.6k
8
votes
Accepted
A question regarding simultaneous congruences
The size of the exceptional set is $\Omega(p^2)$ and indeed, there are at most $\frac6{\pi^2}(1+o(1))p^2$ pairs, projectively equivalent to a point inside the square $Q:=[1,N]\times[1,N]$, where $N=\l …
- 22.6k
12
votes
On Euler's polynomial $x^2+x+41$
Here is a completely elementary and very simple proof.
With the identity $P(A+m^2-1)=P(m-1)P(m)$ in mind, it suffices to prove that if$P(x):=x^2+x+A$ is prime for all $x\in[0,A-2]$, while $P(A+k)$ i …
- 22.6k
1
vote
Size of smallest multiple $\sum \epsilon_i p^i$ of $n$ with digits $\epsilon_i\in \{0,1\}$ f...
While $M=n^{O_p(1)}$ can be tricky (or out of reach), the following argument shows that $M\le p^{(2+o(1))\sqrt{dn}}$ with $d=\gcd(p-1,n)$; thus, $M<\exp(O(\sqrt n))$ in the regime where $p$ is fixed a …
- 22.6k
2
votes
Numbers of the form $2^ma + 2^nb$ where $\text{gcd}(a,b) = 1$
Not necessarily: consider the situation where $a=b=1$ and $d$ is a Mersenne prime.
- 22.6k
4
votes
Accepted
Quadratic character of factorials
In view of the identity $(n!)^2/((n-1)!)^2=n^2$, the set $S_p$ generates the subgroup $Q_p<\mathbb F_p^\times$ of quadratic residues; thus, if $S_p$ is a subgroup, then in fact $S_p=Q_p$. Clearly, a n …
- 22.6k
4
votes
Product of arbitrary Mersenne numbers
Write $p=2^m-1$ and $q=2^n-1$ where $m>n$ without loss of generality. If $pq-1=(2^m-2^{m-n}-1)2^n$ is a square, then so is $2^m-2^{m-n}-1$. But then $m=n+1$, as $m\ge n+2$ implies $2^m-2^{m-n}-1\equiv …
- 22.6k
1
vote
triple with large LCM
For two numbers, this also follows easily from Graham's conjecture. Suppose, to simplify the life, that $n=2k$, and let $a_1>\dotsb>a_k$ be the $k$ largest elements of the set under consideration. By …
- 22.6k
1
vote
triple with large LCM
Here is a direction to explore. I describe it for quadruples, but, perhaps, a similar game can be played with triples.
Suppose that $A$ is a set of $n$ integers, all larger than $n$, such that for an …
- 22.6k