Arithmetic progressions

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In summary, the question is how to show that the arithmetic progression ax+b contains an infinite subsequence, where every two elements are relatively prime. The Chinese Remainder Theorem may be related, but its application is unclear. A counterexample was given for a=2, b=2 when a and b are not coprime, and it was mentioned that the problem actually requires (a,b)=1. The application of the CRT and finding a solution without using Dirichlet's Theorem is still unclear.
  • #1
robin_vanp
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a question came up

"show that the arithmetic progression ax+b contains an infinite subsequence (not necessarily a progression), every two of whose elements are relatively prime."

i have a hunch that the chinese remainder theorem has something to do with this, but I'm not sure how. any thoughts?
 
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  • #2
Is that true? What if a=2, b=o?
 
  • #3
sorry, assuming a, b are non zero
 
  • #4
Then a=2, b=2 is a counterexample. I think you really need that a and b are coprime, in which case the sequence actually contains infinitely many primes.
 
  • #5
right again. its actually a two part question so it says on the top that (a,b) = 1, i forget to mention; if so (now that we finally got the problem) how is the CRT applicable here?
 
  • #6
and deriving some sort of solution that does not employ dirichlet's theorem, i think, because then that would be obvious; i really do not know how the CRT can be used here.
 

1. What is an arithmetic progression?

An arithmetic progression is a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference (d). The general form of an arithmetic progression is a, a+d, a+2d, a+3d, ... where 'a' is the first term of the sequence.

2. What is the formula to find the nth term of an arithmetic progression?

The formula to find the nth term of an arithmetic progression is: a + (n-1)d, where 'a' is the first term and 'd' is the common difference. This formula is also known as the general term formula or the nth term formula.

3. How do you find the sum of an arithmetic progression?

The sum of an arithmetic progression can be found using the formula: S = (n/2)(2a + (n-1)d), where 'n' is the number of terms in the sequence, 'a' is the first term, and 'd' is the common difference. This formula is also known as the sum formula or the arithmetic series formula.

4. How can you determine if a given sequence is an arithmetic progression?

To determine if a given sequence is an arithmetic progression, you can check if the difference between any two consecutive terms is constant. If the difference is constant, then the sequence is an arithmetic progression. You can also use the general term formula to find the next term in the sequence and check if it follows the pattern.

5. What are some real-life applications of arithmetic progressions?

Arithmetic progressions are used in various fields such as finance, physics, and computer science. In finance, they are used to calculate compound interest and depreciation. In physics, they are used to model motion with constant acceleration. In computer science, they are used in algorithms and data structures, such as binary search and hash tables.

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