Combinatorial Number Theory Problem

In summary: I think this is a case of too much hay and too little needle. In summary, the problem presented involves sets A and B of natural numbers, where the pairwise sums of each number in A and B are equal, but the cardinalities of A and B are not necessarily equal. It is suggested to use induction on the sum of the cardinalities of A and B, but further restrictions may be needed to make the problem solvable.
  • #1
Mathguy15
68
0
Hello,

I would like to see a solution to the following problem:

Let A be a finite collection of natural numbers. Consider the set of the pairwise sums of each of the numbers in A, which I will denote by S(A). For example, if A={2,3,4}, then S(A)={5,6,7}. Prove that if S(A)=S(B) for two different finite sets of natural numbers A and B, then |A|=|B|, and |A|=|B| is a power of 2.

I find this problem interesting, but I am working on other problems. Anyone have ideas?

Thanks,
Mathguy
 
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  • #2
You haven't defined |A| for this situation.
 
  • #3
For a set C, |C|=the cardinality of C.
 
  • #4
Mathguy15 said:
Hello,

I would like to see a solution to the following problem:

Let A be a finite collection of natural numbers. Consider the set of the pairwise sums of each of the numbers in A, which I will denote by S(A). For example, if A={2,3,4}, then S(A)={5,6,7}. Prove that if S(A)=S(B) for two different finite sets of natural numbers A and B, then |A|=|B|, and |A|=|B| is a power of 2.

I find this problem interesting, but I am working on other problems. Anyone have ideas?

Thanks,
Mathguy



Well, if one considers zero a natural number (many do) then both claims above are false, as [itex]\{2,3,4\},\{0,2,3,4\}[/itex] , otherwise...perhaps by induction on [itex]|A|+|B|[/itex] ...

DonAntonio
 
  • #5
Maybe I'm missing something, but it looks like for A = {1,2,3,4,5,6,7} and B = {1,2,3,5,6,7}

S(A) = {3,4, ... 12,13} = S(B), but |A|=7 and |B|=6

Maybe there should be some other restrictions on A and B?

(guessing) If neither A nor B may be a subset of the other, then consider {1,2,3,4,6,7,8,9}, {1,2,3,5,7,8,9} . Cardinalities 8 and 7.
 
  • #6
Great Scott, you may be right catellus!
 

1. What is Combinatorial Number Theory Problem?

Combinatorial Number Theory Problem is a branch of mathematics that combines elements of combinatorics and number theory to solve problems related to counting, arrangements, and combinations of numbers. It involves the study of integers and their properties, along with the application of techniques from discrete mathematics to solve problems.

2. What are some applications of Combinatorial Number Theory Problem?

Combinatorial Number Theory Problem has various applications in fields such as computer science, cryptography, and physics. It is used to solve problems related to coding theory, graph theory, and data encryption. It also has applications in the study of prime numbers, partitions, and permutations.

3. What are the main techniques used in Combinatorial Number Theory Problem?

The main techniques used in Combinatorial Number Theory Problem include generating functions, combinatorial identities, and number theoretic techniques such as modular arithmetic and prime factorization. Other techniques like inclusion-exclusion principle and pigeonhole principle are also commonly used.

4. How is Combinatorial Number Theory Problem connected to other branches of mathematics?

Combinatorial Number Theory Problem is closely related to other branches of mathematics such as algebra, geometry, and analysis. It uses techniques from these fields to solve problems related to counting and arranging numbers. It also has connections to topics like group theory, probability, and discrete optimization.

5. What are some well-known problems in Combinatorial Number Theory?

Some well-known problems in Combinatorial Number Theory include the Goldbach conjecture, the twin prime conjecture, and the Collatz conjecture. Other famous problems include the Riemann hypothesis, the four color theorem, and the perfect graph conjecture. These problems have been studied by mathematicians for decades and continue to be open areas of research.

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