Highest Common Factor to Infinity

Question

Imagine you have a set of integers of x.For example:

7 9 11 13

Let us imagine that y is 1.

Then for each nth generation you added 1 to each member of the set, found the HCF of the set and set y to the HCF if it was bigger than y originally.

For example, when n was 9, the set would be:

16 18 20 22

and the HCF of the set would be 2.

Would there be a limit to y (the biggest HCF found) when n became infinity?

Answer 1

Let set $X$ be arranged in increasing order, and have elements $ x_1 < x_2 < \ldots < x_k$.

You are asking for

$$ \max_n HCF ( n + x_1, n+x_2, \ldots n+x_k)$$

Claim: This is equivalent to

$$ \max_n HCF ( n + x_1 , x_2 - x_1, x_3 - x_1, \ldots , x_k - x_1 )$$

Claim: This is equivalent to

$$ HCF ( x_2 - x_1, x_3 - x_1, \ldots , x_k - x_1 ).$$

Hence, there is a maximum, which is easily described.

Can you fill in the proofs of the claims yourself?