All Questions
7 questions
2
votes
1
answer
513
views
Prove ${^{b}a} \equiv {^{b+1}} a \pmod {10^{\lfloor{\log_{10} (^{b}a) }\rfloor + 1}} \Rightarrow a=5$ as $a$ and $b$ are two integers greater than $1$
$\DeclareMathOperator\len{len}$Let $a, b \in \mathbb{N} -\{ 0, 1 \}$ and define ${^{b}a}$ to be $a^a$ if $b = 2$ and $a^{\left(^{b-1}a \right)}$ if $b \geq 3$ (e.g., ${^{3}5} = 5^{\left( 5^5 \right)} =...
- 1,451
6
votes
2
answers
646
views
Number of solutions of $am \equiv bn \pmod{q}$
Let $q$ be a (large) prime. Let $N$ be a positive integer of size${}\approx \sqrt{q}$. Let $\mathcal{M}$ be an arbitrary subset of $\{1, \dots, q\},$ such that $\mathcal{M}$ has cardinality $N$. ...
- 2,207
0
votes
0
answers
132
views
Which consequences can be deduced from this peculiar property of tetration?
Recently (assuming radix-$10$), I showed that, for any $a \in \mathbb{N}_{0}$ that is not a multiple of $10$, there exists a unique value $V(a) \in \mathbb{N}_{0}$ which corresponds to the number of ...
- 1,451
4
votes
2
answers
202
views
System of congruences with bound condition
Let $m$ be a positive integer divisible by $6$, and let $q$ be one of $8,9$,
or a prime $\gt 3$.
Question : Is there always an $x\in [1,m]$, coprime to $m$, such
that $x\not\equiv\pm 1 \ \mod{q}$ ?
...
- 621
7
votes
0
answers
183
views
Some conjectural congruences involving Domb numbers
The Domb numbers are given by
$$D_n=\sum_{k=0}^n\binom{n}{k}^2\binom{2k}k\binom{2(n-k)}{n-k}\ \ \ (n=0,1,2,\ldots).$$
Such numbers have combinatorial interpretation, see, e.g., http://oeis.org/A002895....
- 15.6k
6
votes
0
answers
217
views
Two conjectural congruences for Franel numbers
Recall that the Franel numbers are given by
$$f_n:=\sum_{k=0}^n \binom{n}{k}^3\ \ \ (n=0,1,\ldots).$$
Question. How to prove my following conjecture?
Conjecture. For each odd prime $p$, we have
$$\...
- 15.6k
3
votes
1
answer
314
views
Probability in $GL_2(\mathbb{Z}/p^{r}\mathbb{Z})$
My question may be not interesting or easy to answer ! but I am really not familiar with proba.
Let $p$ be an odd prime number. and let $r\geq1$ an integer. choose an element $A\in\mathrm{GL}_2(\...
- 347