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Results tagged with nt.number-theory
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user 9025
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
1
vote
how to find cubic polynomial that an unknown subset of a set of integers satisfies
I don't see how this can be done exhaustively for a large number of integers, but I can suggest an approach that may be useful in practice. First sort your $N$ intergers as $x_1\lt x_2\lt\cdots\lt x_ …
- 37k
3
votes
Defining $\{a_i\}$ as $(1+x+⋯+x^k)^n =\sum_{i=0}^{kn}a_ix^i$, then is the 'special' differen...
Yes, because convolutions of log-concave sequences are log-concave. Products of polynomials are convolutions of their coefficient sequence. Search on these keywords and you'll find tons of references …
- 37k
4
votes
Reference : Partition of integer
It is http://oeis.org/A006171 , which has some generating functions and other formulas.
- 37k
2
votes
Particular complex fractions
This is well known, but I'll give a short proof using 3 dimensions as an example.
Every power $x^n$ can be written as the integer linear combination of binomial coefficients $\binom xj$ for $0\le j\le …
- 37k
4
votes
How many 0, 1 solutions would this system of underdetermined linear equations have?
As other people noted, this is a #P-hard problem and you cannot hope to count the solutions in time which is polynomial in the size of the problem. However, in many cases you can do it a lot faster t …
- 37k
2
votes
Asymptotic equivalence for functions with zeros
It isn't clear if you intend that $f$ and $g$ are eventually zero at the same places. Otherwise I wouldn't want to call them asymptotically equivalent. What you need is
$$ f(x) = (1+o(1)) g(x), $$
w …
- 37k
2
votes
Is this variant on set partition explored?
The number can be expressed as a sum, though it isn't too enlightening. Let $m$ be the number of cells of the third type (there exists $i,j\in B$ such that $i\lt r\lt j$). Let $k_A$ be the number of e …
- 37k
9
votes
Accelerating convergence for some double sums
Here is a little Maple. Note that using "sum" on the inside causes it to find an algebraic expression for the sum over $\ell$ and using "Sum" on the outside tells it to not try to sum that algebraica …
- 37k
4
votes
Mathematical techniques to reduce the amount of storage memory
It seems like you want to use the same number of bits for each customer. That is a big mistake and you won't find a good solution unless you drop that requirement.
What you need is an adaptive schem …
- 37k
14
votes
Are (55, 165, 495, 1485) and (286, 1716, 10296, 61776) the only geometric sequences of lengt...
(Partial results.)
For the case of integer ratio, there are only two sequences of 4 binomials in geometric progression for which the largest is at most $10^{17}$. Namely, 55,165,495,1485 found by Will …
- 37k
5
votes
Closed formula for number of ones in a proper factor tree
Take the formal product $g(x_1,x_2,\ldots)=\prod_{i\ge 1} (1-x_i)$ and
define
$$f(x_1,x_2,\ldots) = \frac{g(x_1,x_2,\ldots)}{2g(x_1,x_2,\ldots)-1}.$$
Then
$\gamma(\prod_i p_i^{\alpha_i})$ is the coeff …
- 37k
10
votes
Is it possible to stab (every rotation of) any four element subset of $\mathbb Z_n$ with les...
NEW VERSION: (What was I thinking?)
A greedy algorithm gives a stronger result.
THEOREM. Consider any family $\mathcal F$ of $n$ 4-subsets of $\lbrace 1,\ldots,n\rbrace$. Then there is a set $X\sub …
- 37k
1
vote
Expectation of edge weights on the complete graph
(Not a complete solution.)
An interesting property is this: For an edge $uv$, the distribution of $b(u)+b(v)$ conditioned on $b(u)$ is the same as the unconditional distribution (namely uniform). From …
- 37k
2
votes
Accepted
Does this quadratic system admit an integral or a rational solution?
In my comments I employed Maple, which uses tools like Grobner bases to solve polynomial equations. But now I'll try to do it by hand. Let $E_1,E_2,E_3$ be the three equations. A rational solution of …
- 37k
2
votes
Non-singular matrix with restricted entries
[EXPANDED]
PART 1 (also done by Peter)
If $x,y$ are coprime and have opposite sign, there is a singular symmetric matrix with 1 on the diagonal and
only $x$ and $y$ off the diagonal.
Say $x<0,y>0$. Si …
- 37k