We have recurrent sequence of integer number a_{1},a_{2},...
a1=1, a2=2
a_{n}=3a_{n-1}+5a_{n-2} for n=3,4,5,...
Is integer number k>=2, that (a_{k+1}*a_{k+2}) mod a_{k} = 0 ?
We have recurrent sequence of integer number a_{1},a_{2},...
a1=1, a2=2
a_{n}=3a_{n-1}+5a_{n-2} for n=3,4,5,...
Is integer number k>=2, that (a_{k+1}*a_{k+2}) mod a_{k} = 0 ?
Please for quick help :)
You need to clarify your post A(2) = 2. A(3)*A(4) = 11*43 is not divisible by A(2). Do you mean to ask whether for some integer n that a_{n}|a_{n+1}*a_{n+2}?
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#3
terafull
3
0
Yes. I must find some integer n (exactly k), where this modulo statement is true.
Thanks for reply :)
#4
dodo
695
2
I was doing some number crunching to reduce the possibilities for k, but I'm still far from an answer.
So far, I get the following:
If b, a, 3a+5b, 14a+15b, ... are contiguous elements of the sequence, then we see that, if a_n=b is even, then a_{n+3}=14a+15b is also even. And since there happens to be an even element among the first 3 (namely, a_2=2, then one every 3 elements from that point on (a_5, a_8, a_{11}...) will be even too.
In short, since a_2=2,
n \equiv 2 \ (mod\ 3) \ \ \Rightarrow \ \ 2|a_n
Which means that the desired k cannot be congruent to 2 (mod 3), because a_k would have a factor 2 that (a_{k+1} * a_{k+2}) doesn't have.
and given that a_3=11, a_4=43, a_5=23\ .\ 8, a_6=13\ .\ 59 \ \mbox{and}\ a_8=17\ .\ 794, we also have
n \equiv 3 \ (mod\ 12) \ \ \Rightarrow \ \ 11|a_n
n \equiv 4 \ (mod\ 11) \ \ \Rightarrow \ \ 43|a_n
n \equiv 5 \ (mod\ 11) \ \ \Rightarrow \ \ 23|a_n
n \equiv 6 \ (mod\ 12) \ \ \Rightarrow \ \ 13|a_n
n \equiv 8 \ (mod\ 18) \ \ \Rightarrow \ \ 17|a_n
and all these conditions (including the one above about even numbers) must be avoided by your k candidate. However, there are still plenty of valid candidates remaining.
I asked online questions about Proposition 2.1.1:
The answer I got is the following:
I have some questions about the answer I got.
When the person answering says:
##1.##
Is the map ##\mathfrak{q}\mapsto \mathfrak{q} A _\mathfrak{p}## from ##A\setminus \mathfrak{p}\to A_\mathfrak{p}##?
But I don't understand what the author meant for the rest of the sentence in mathematical notation:
##2.##
In the next statement where the author says:
How is ##A\to...
The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho.
In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states
"Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels).
Also in the reference url page above, the authors present two...
##\textbf{Exercise 10}:##
I came across the following solution online:
Questions:
1. When the author states in "that ring (not sure if he is referring to ##R## or ##R/\mathfrak{p}##, but I am guessing the later) ##x_n x_{n+1}=0## for all odd $n$ and ##x_{n+1}## is invertible, so that ##x_n=0##"
2. How does ##x_nx_{n+1}=0## implies that ##x_{n+1}## is invertible and ##x_n=0##. I mean if the quotient ring ##R/\mathfrak{p}## is an integral domain, and ##x_{n+1}## is invertible then...
