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.

#NumberTheory #Generalize #Divisibility #infinite #Existence

Easy Math Editor

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`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}`	$\boxed{123}$

Comments

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,

Abhishek Sinha - 6 years, 11 months ago

Hint: $a_1 = b_1$

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

Ivan Koswara - 6 years, 11 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 $a_1\ne b_1$ .

Daniel Liu - 6 years, 11 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?

Siam Habib - 6 years, 11 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!

Daniel Liu - 6 years, 11 months ago

Math	Appears 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}`	$\boxed{123}$