All Questions
10 questions
4
votes
2
answers
593
views
Squares in Lucas sequences
Good night, everyone!
According to a celebrated result by J. H. Cohn, the only perfect squares in the Fibonacci sequence are $F_{0}=0$, $F_{1}=F_{2}=1$, and $F_{12}=144$. It is also known that the ...
- 87
1
vote
0
answers
108
views
Question related to sequence of recurrence relation $a_k=\operatorname{rad}(a_{k-1}+a_{k-2})$ for $k\ge 2$ where $a_0=0,a_1=1$
Define radical of an integer Wiki
$$\displaystyle{\mathrm{rad}}(n)=\prod_{{\scriptstyle p\mid n\atop p\:{\text{prime}}}}p$$
Example $n=504=2^3\cdot3^2\cdot7$ therefore ${\displaystyle \operatorname{...
- 599
1
vote
0
answers
37
views
Raggedness measure of a sequence
This surely has been done, maybe I googled the wrong adjective...
Define a raggedness measure $r$ of a sequence $S$ in this way:
Two members $S_i,S_j$ of the sequence (who don't have to be adjacent!) ...
- 4,829
9
votes
2
answers
1k
views
The p-adic valuation of a linear recurrence
Let $(u_n)_{n \geq 0}$ be an integer-valued linear recurrence of order $k \geq 1$. Precisely,
$$u_n = a_1 u_{n-1} + \cdots + a_k u_{n - k} \quad \forall n \geq k ,$$
for some $a_1, \ldots, a_k \in \...
user40023
0
votes
0
answers
86
views
Polynomials of integer coefficients that evaluated at Mersenne or Fermat numbers produce square-free integers
Mersenne numbers $M_n=2^n-1$ and Fermat numbers $F_n=2^{2^n}+1$ draw the attention of professional mathematicians to get prime constellations or statements related to primality tests for these ...
17
votes
2
answers
3k
views
Some unpublished notes of Hofstadter
I'm looking for some unpublished notes called "Eta Lore," which are apparently related to a talk Douglas Hofstadter first gave at the Stanford Math Club in 1963. I know these notes exist because they'...
5
votes
0
answers
317
views
Elliptic curve sequences needed for universal forgery
Elliptic Curve Digital Signature Algorithm (ECDSA) admits universal forgery (UF) if the Attacker can solve the equation
$$z=\frac{f_{k-1}(x,y)f_{k+1}(x,y)}{f_{k}(x,y)^2},$$
where $k$ is unknown, $f_{k}...
- 12.3k
2
votes
0
answers
154
views
Equi-distribution of the parity of partitions
The integer partition function $p(n)$ has a generating function given by
$$\frac1{(q)_{\infty}}=\sum_{n=0}^{\infty}p(n)q^n$$
with $(q)_{\infty}=\prod_{m=1}^{\infty}(1-q^m)$. The long-standing problem ...
- 43.2k
1
vote
1
answer
81
views
Reference request for multiple free sequences
Erdos usually named a sequence of integers no one of which is divisible by any other as an $M$- sequence (M stands for "multiple-free") or primitive sequence.
For example it is easy to see that $\...
- 3,876
7
votes
1
answer
283
views
On one class of Somos-like sequences
This question is motivated by integrability of the sequence mistakenly arisen in the question Does this sequence always give an integer?
Let $m_1,\ldots, m_{k-1}$ be positive integers and sequence $\{...
- 12.3k