Proof about arithmetic progressions

  • Thread starter Thread starter cragar
  • Start date Start date
  • Tags Tags
    Arithmetic Proof
Click For Summary

Homework Help Overview

The discussion revolves around proving the existence of arbitrarily long arithmetic progressions of different positive integers where every two terms are relatively prime. The original poster explores various approaches, including the properties of odd integers and factorials, to establish a valid progression.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • The original poster considers using odd integers and their properties, questioning the implications of using consecutive odd integers. They also explore the idea of using factorials in the progression. Other participants raise questions about the relationship between common factors and the step size in the progression.

Discussion Status

Participants are actively engaging with the problem, providing hints and exploring different formulations of the arithmetic progression. Some guidance has been offered regarding the use of factorials, and there is a sense of productive direction as participants refine their approaches.

Contextual Notes

There is an ongoing discussion about the assumptions regarding the nature of the integers in the progression and the implications of using different step sizes. The original poster's attempts indicate a need for clarity on the conditions under which the integers remain relatively prime.

cragar
Messages
2,546
Reaction score
3

Homework Statement


Prove that there exist arbitrarily long arithmetic progressions formed of different positive integers such that every 2 terms of these progressions are relatively prime.

The Attempt at a Solution


First i started to think about the odd integers like 2x+1 and how consecutive odd integers are relatively prime because their difference is 2 , but I am pretty sure the question is asking about any 2 terms in the progression. Then i started to think about progressions of odd numbers that were separated by powers of 2 but then this would be a progression because the difference between consecutive numbers would not be the same. Then I was trying to think of something i could do with a factorial.
 
Physics news on Phys.org
Deleted - sorry, I wasn't thinking straight.
 
Last edited:
If two terms have a common factor, p, what can you say about their difference? If the step size is n!, what can you say about the relationship between p and n?
 
ok thanks for the hint haruspex, I think n!x+1 will work where x=1,2,3... and n is a natural number as large as you like. if we look at the difference between any 2 of the terms in the progression.
like n!(x+a)+1-(n!x+1) =n!a where n>x+a , if n!(x+a)+1 and n!x+1 had common factors then it should divide their difference but their difference is n!a, and none of those factors divide n!x+1 because their would be a remainder because n! contains all the factors 1 up to n.
 
cragar said:
ok thanks for the hint haruspex, I think n!x+1 will work where x=1,2,3... and n is a natural number as large as you like. if we look at the difference between any 2 of the terms in the progression.
like n!(x+a)+1-(n!x+1) =n!a where n>x+a , if n!(x+a)+1 and n!x+1 had common factors then it should divide their difference but their difference is n!a, and none of those factors divide n!x+1 because their would be a remainder because n! contains all the factors 1 up to n.
Glad that worked - seemed like it should but I hadn't checked the details.
 

Similar threads

Proof about arithmetic progressions
  • · Replies 4 ·
Replies
4
Views
2K
Solving for nC8,nC9, and nC10 in an Arithmetic Progression
  • · Replies 3 ·
Replies
3
Views
2K
Arithmetic progression topology, Z not compact
  • · Replies 16 ·
Replies
16
Views
4K
Graduate Is this Recursive Pattern in the Collatz Sequence Known?
  • · Replies 3 ·
Replies
3
Views
1K
Proof about relatively prime integers.
  • · Replies 1 ·
Replies
1
Views
2K
Terms of a geometric series and arithmetic series, find common ratio
  • · Replies 3 ·
Replies
3
Views
3K
MHB Arithmetic Progression: Expressing d in Terms of x,y,z,n
  • · Replies 1 ·
Replies
1
Views
1K
Linear Subspace of R^n: Arithmetic Progressions Verification
  • · Replies 2 ·
Replies
2
Views
2K
Assume that if the real-valued function h(x) is Lipschitz continuous...
  • · Replies 22 ·
Replies
22
Views
2K
Relatively prime proof involving a^n and b^n
  • · Replies 3 ·
Replies
3
Views
2K