Consecutive integers and relatively prime numbers

In summary, consecutive integers are a sequence of numbers that follow each other in order with a difference of 1 between each number. Relatively prime numbers are two numbers that have no common factors other than 1. Consecutive integers cannot be relatively prime because they always have a common factor of 1. To determine if two consecutive integers are relatively prime, you can find their greatest common divisor. There are special properties of consecutive integers and relatively prime numbers, such as any two consecutive integers are always relatively prime and the product of any two consecutive integers is also relatively prime to either of the numbers.
  • #1
e2m2a
354
11
Summary:: Interested in the history of the proof.

Consecutive integer numbers are always relatively prime to each other. Does anyone know when this was proved? Was this known since Euclid's time or was this proved in modern times?
 
Physics news on Phys.org
  • #2
I assume this has been known since someone observed divisors. It is so elementary that it doesn't even deserve the label 'proof': ##d|a \wedge d|(a+1) \Longrightarrow d|((a+1)-a)=1 \Longrightarrow d\in \{\pm 1\}.##

I don't think there is a book History Of Proofs. The earliest proofs are probably the geometry of the ancient Greeks.
 

1. What are consecutive integers?

Consecutive integers are a sequence of numbers that follow each other in order without skipping any numbers. For example, 1, 2, 3, 4, 5 are consecutive integers.

2. How do you determine if two numbers are relatively prime?

Two numbers are relatively prime if they have no common factors other than 1. This means that their greatest common divisor (GCD) is 1. One way to determine this is by finding the prime factorization of both numbers and checking if they have any common factors.

3. Can consecutive integers be relatively prime?

Yes, consecutive integers can be relatively prime. For example, 3 and 4 are consecutive integers and they are relatively prime because their only common factor is 1.

4. What is the relationship between consecutive integers and relatively prime numbers?

There is no direct relationship between consecutive integers and relatively prime numbers. However, it is possible for consecutive integers to be relatively prime, as mentioned in the previous question.

5. How can you use consecutive integers to find relatively prime numbers?

To find relatively prime numbers using consecutive integers, you can start with any two consecutive integers and check if they are relatively prime. If not, you can add or subtract 1 from one of the numbers and check again until you find a pair of relatively prime numbers.

Similar threads

Replies
1
Views
773
Replies
1
Views
1K
  • Precalculus Mathematics Homework Help
Replies
9
Views
1K
  • Precalculus Mathematics Homework Help
Replies
3
Views
947
  • Engineering and Comp Sci Homework Help
Replies
32
Views
3K
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
7
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
748
  • Calculus and Beyond Homework Help
Replies
9
Views
1K
Replies
3
Views
1K
Back
Top