Peter Breuer Thank you Dr. Peter Breuer,

A set of n consecutive positive integers is said to have a relatively prime element (Shortly relprime) if there is an element in that set which is relatively prime to the others (the rest of the numbers). The set {2184, 2185, ..., 2200} has no relprime. Let A be a set of consecutive positive integers. An element α of A is called an s-near relative prime number (s-near relprime in short) if α is coprime to any distinct element in A except exactly s elements of A. With best regards

The set {4,5,6,7,8,9,10} has a relprime which is 7, since 7 is relatively prime to all other elements. Also 9 is 1-near replime, because it is relatively prime to all other elements except 6.

I wonder if there is a set of n consecutive positive integers such that this set has no relprime (like the set [2184, 2200] ) and has no 1-near relprime. I conjecture that such a set does not exist

Any sequence of n consecutive positive integers possesses a relprime if n is less than or equal 16 (Pilliai's Theorem). If n is greater than or equal 17, many sequences have no relprime.

All of the following sets have no relprime:

[2184,2200],

[27830,27846],

[27829,27846],

[27828,27846]. The first set has 1-near relprime, which is either 2189 or 2197. All the other sets have 1-near relprime.

Upgrade the question. Is there a sequence of consecutive positive integers A possess the following properties.

1. A has no relprime.

2. A has no 1-near relprime.

3. A has 2-near relprime.

Peter Breuer. Thank you for your concerns.

You seem to have difficulty in saying "yes" or "no"! (YES).

Given a set A={k,k+1, ..., k+n-1} of n positive integers. I want to find a specific set A such that A satisfies the three properties mentioned in my last reply. The set A will contain at least 17 elements. The length of shortest sequence that has no relprime is 17.

Why the change? Have you found an answer for the earlier version?(NO)