Divisibility Chains Generalization

Based on this problem.


Define two positive integer sequences {an}\{a_n\} and {bn}\{b_n\} be defined as a1b1a_1\ne b_1, an+1=an+ka_{n+1}=a_n+k and bn+1=bn+kb_{n+1}=b_n+k. These two sequences form an Order kk Divisibility Chain of length nn if aibia_i\mid b_i for i=1ni=1\to n.

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


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
5 years, 2 months ago

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Take a1=k,b1=k+k×n!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 - 5 years, 2 months ago

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Hint: a1=b1a_1 = b_1

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

Ivan Koswara - 5 years, 2 months ago

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I can't believe I didn't realize that any a1=b1a_1=b_1 works. Edited the problem and this note to show that a1b1a_1\ne b_1.

Daniel Liu - 5 years, 2 months ago

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Hint: Consider the sequence in modulo lcm(a1,a2,...,an)lcm(a_1,a_2,...,a_n). From there rest will be pretty easy.

@Daniel Liu : Am I being too obvious?

Siam Habib - 5 years, 2 months ago

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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 - 5 years, 2 months ago

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