0
votes
1answer
36 views

Sequence That increases and then decreases using Modular Arithmetic

I'm trying to find a simple formula for a periodic sequence like this: $$ 0,1,2,3,4,3,2,1,0...$$ I've figured it out for the increasing part of the sequence by using a modulo operator: $$ i\ mod \ ...
3
votes
1answer
62 views

Sequence becomes constant modulo $n$

Does the sequence $$a,a^a,a^{a^a},\cdots$$ is constant modulo $n$ from a certain rank ? Where $a,n \in \mathbb{N}$ Using mathematica I am tempted to say yes but I how can I approach this ? ...
1
vote
2answers
42 views

$a_3$, $a_5$ and $a_0$ terms are required?

We have an arithmetic sequence $a_n>0$ and it's increasing. and we've two systems of equations: $a_4=15$, $m+d=21$ whereas $m=lcm(a_3,a_5)$, $d=\gcd(a_3,a_5)$. What are the values of $a_3$, ...
41
votes
5answers
4k views

Does every prime divide some Fibonacci number?

I am tring to show that $\forall a \in \Bbb P\; \exists n\in\Bbb N : a|F_n$, where $F$ is the fibonacci sequence defined as $\{F_n\}:F_0 = 0, F_1 = 1, F_n = F_{n-1} + F_{n-2}$ $(n=2,3,...)$. How can ...
1
vote
1answer
63 views

Sum of a series involving modulus operator

I'm attempting to work out a problem that involves summing a series of numbers. I know the formula to find each element of the series, but I do not know how to use this to make an equation for the ...
0
votes
1answer
59 views

A result about quadratic numbers

I'm stuck in the middle of an exercise about quadratic numbers. Let me quickly sum it up. Let $d$ be a positive integer that is not the square of any integer. Let $n \in \mathbb N^*.$ Prove the ...
4
votes
1answer
88 views

$m \in \{2,6,42,1806,…\} $ - a problem of sum-of-$m$'th powers modulo $m$

(continuing the work for an answer for a question here in MSE and also in MO) I'm (re-)viewing the function $$ f(m) = \sum_{k=0}^{m-1} k^m $$ considering its residue modulo $m$: $$ r(m) \equiv f(m) ...
-4
votes
2answers
563 views

How many pairs of integers $(A, B)$ are there in the range $[1,\ldots, N]$, such that $\gcd(A,B) = B$?

I am given a positive integer $N$ ($N\leq 10^9$). How many pairs of integers $(A, B)$ exist in the range $[1,\ldots, N]$ such that $\gcd(A,B) = B$?
2
votes
1answer
55 views

Finding Similar Sequences

Can we find two sequences: $$\{a (b^0), a (b^1), a (b^2), a (b^3), \dots, a(b^n)\} \bmod p_1$$ $$\{c (d^0), c (d^1), c (d^2), c (d^3), \dots, c(d^n)\} \bmod p_2$$ that differ by only one number? ...
5
votes
3answers
84 views

Evaluate $\sum\limits_{(m,n) \in D,m < n} \frac{1}{n^2 m^2} $ where $\gcd(m,n)=1$

i have no clue on how to evaluate: $$\sum\limits_{(m,n) \in D,m < n} \frac{1}{n^2 m^2} \text{ where }D = \{ (m,n) \in (\mathbb{N}^*)^2 \mid \gcd(m,n) = 1\} $$ If someone is able to give me a ...
1
vote
1answer
159 views

Homework problem on identifying a sequence

I had this problem in my discrete math/modular arithmatic course where I had to find the first 10 terms of a series F(r), starting from F(3). The given information is: F(3)=1 F(4)=13 F(10) % ...
1
vote
1answer
50 views

REVISTED$^1$ - Order: Modular Arithmetic

Relevant Literature: Question: Observe that $2^{10}=1024≡−1 \pmod{25}$.Find the order of $2$ modulo $25$. Thoughts: Direct answers are OK, but I'd like to know if I'm right that what I'm really ...
1
vote
1answer
48 views

Formula for working out an ID number by given set of coordinates

I'm designing an online game and having a bit of a mental block coding the navigation system. It's designed on a 2 dimensional grid, each cell has an ID 0...n, n being the total number of cells in the ...
2
votes
0answers
29 views

Name for summation that returns GCD(k,n)

Define $$ H(k,n)=2\sum_{i=1}^{n-1}\left\lfloor\frac{ki}{n}\right\rfloor\;. $$ We can prove that $H(k,n)=nk-k-n+\gcd(k,n)$. Does this $H$ carry some known name?
0
votes
0answers
45 views

Minimum of a linear congruence sub-sequence

I have the following little problem : let $a,b,u,v$ be four given integers with $\gcd(a,b)=1$. Now I would like to find the minimum of the linear congruence subsequence $\{ax \pmod b : u \le x \le ...
1
vote
0answers
104 views

Cycle of remainders

Let $N, K, W$ be natural numbers If I start from $R_0$, say any integer $r_0, 0 \lt r_0 \lt N$ and proceed with: $$R_j = ( R_{j-1} + K ) \mod W,\quad j=1,2, \dots$$ (that is the remainder of the ...
1
vote
1answer
300 views

How to remove the denominator?

I have the following expression for $n>3$: $$\frac{5\cdot(n-1)\cdot[8\cdot\operatorname{Luc}(n) + 5\cdot\operatorname{Luc}(n-1)] + [4\cdot\operatorname{Luc}(n-1) - ...
1
vote
3answers
193 views

Summing a function using modulus

The problem: If the infinite sum of a function is known, how to find: $$\begin{align*} \sum_{i\equiv 0 \mod m}f(x_0+i)=\\ f(x_0)+f(x_0+m)+f(x_0+2m)+f(x_0+3m)+\ldots \end{align*}$$ And if the ...
0
votes
0answers
98 views

Modular sequences

How might one show that if given that there exist some integer $x_0$ s.t. ${x_0}^2 \equiv c \pmod p$ for some integer $c$ that is not a square and not a multiple of $p$, then there exist $x_n$ s.t. ...
2
votes
1answer
445 views

Prove that a given sequence is periodic modulo m

How can one tell that a given sequence is perodic modulo m? For example its easy to see that the sequence $1^1, 2^2, 3^3, \dots$ is periodic modulo 10. But how can we prove this?
0
votes
1answer
179 views

Why does this sum mod out to 0?

In making up another problem today I came across something odd. I've been thinking it over and I can't exactly place why it's true, but after running a long Python script to check, I haven't yet found ...