All Questions
Tagged with combinatorics number-theory
1,985
questions
5
votes
1
answer
3k
views
Pairs of numbers with a given LCM
How can we show that the number of pairs $(a,b)$ (where the pairs $(a, b)$ and $(b, a)$ are considered same) with $\operatorname{lcm}(a, b) = n$ is equal to
$$\displaystyle\frac{(2e_1+1)(2e_2+1)...(...
- 589
6
votes
3
answers
358
views
Constructing a seven digit number such that one divides the other
This was asked to me by a friend. I tried this problem for a long time, but i couldn't solve it. Seems interesting though.
Prove that it is impossible to construct two different seven digit numbers, ...
anonymous
60
votes
1
answer
4k
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Quadratic reciprocity via generalized Fibonacci numbers?
This is a pet idea of mine which I thought I'd share. Fix a prime $q$ congruent to $1 \bmod 4$ and define a sequence $F_n$ by $F_0 = 0, F_1 = 1$, and
$\displaystyle F_{n+2} = F_{n+1} + \frac{q-1}{4} ...
- 432k
6
votes
1
answer
262
views
Recreating an Integer Sequence After Convolution
...and encoding it as a probability distribution.
Suppose we have a sequence of non-negative integers that is periodic with period $N$:
\begin{equation*}
A_{1},A_{2},...,A_{N},A_{1}...
\end{...
- 1,086
44
votes
3
answers
10k
views
How do I count the subsets of a set whose number of elements is divisible by 3? 4?
Let $S$ be a set of size $n$. There is an easy way to count the number of subsets with an even number of elements. Algebraically, it comes from the fact that
$\displaystyle \sum_{k=0}^{n} {n \...
- 432k