Divisibility Chains Generalization

Based on this problem.

Define two positive integer sequences $$\{a_n\}$$ and $$\{b_n\}$$ be defined as $$a_1\ne b_1$$, $$a_{n+1}=a_n+k$$ and $$b_{n+1}=b_n+k$$. These two sequences form an Order $$k$$ Divisibility Chain of length $$n$$ if $$a_i\mid b_i$$ for $$i=1\to n$$.

Prove that no matter what $$n$$ and $$k$$ you choose, there always exists an infinite number of sequences $$\{a_n\}$$ and $$\{b_n\}$$ that form an Order $$k$$ Divisibility Chain of length $$n$$.

Please only post hints, do not post the solution. If you do, it will give away the solution for the problem I based this on. Thanks.

Note by Daniel Liu
4 years, 4 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

Sort by:

Take $$a_1=k, b_1=k+k\times n!$$. This gives an example of a single such sequence for any n,k. Should not be hard to generalize this to produce infinitely many such sequences,

- 4 years, 4 months ago

Hint: Consider the sequence in modulo $$lcm(a_1,a_2,...,a_n)$$. From there rest will be pretty easy.

@Daniel Liu : Am I being too obvious?

- 4 years, 4 months ago

No, you're not being too obvious as far as I can tell.

Now to think about it, I should have posted this problem for the Proofathon Sequences and Series competition. Dang!

- 4 years, 4 months ago

Hint: $$a_1 = b_1$$

(Which, by the way, is why I reported the problem.)

- 4 years, 4 months ago

I can't believe I didn't realize that any $$a_1=b_1$$ works. Edited the problem and this note to show that $$a_1\ne b_1$$.

- 4 years, 4 months ago

×