Fundamental Theorem of Arithmetic Problem

In summary, The conversation discusses problem 4 of a given image and how to solve it. The participants discuss the importance of paying attention to the fact that t1, t2, etc. are greater than or equal to zero. They also talk about the problem of multiplying exponents and the need for a formal proof to show the uniqueness of prime factorization. Some participants suggest referring to "Principia Mathematica" for a formal proof, while others mention using problem 3 as a starting point but also acknowledging the challenges of 0 being a prime factor. Ultimately, the group agrees that the key to solving problem 4 is understanding how to express "a" and "b" as a product of common primes to a non-negative
  • #1
Fisicks
85
0
Problem number 4 on the image has me stumped. I understand the problem (obviously not enough) and what its saying, I'm just having trouble putting it into a proof. Can i get a hint to get me started? Thanks

http://img40.imageshack.us/i/asdasdjql.jpg/" [Broken]
 
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  • #2
I think you should pay attention to the fact t1 , t2, etc . are greater than OR EQUAL TO zero.
This should solve your purpose.
 
  • #3
srijithju said:
I think you should pay attention to the fact t1 , t2, etc . are greater than OR EQUAL TO zero.

I think you mean g_i and h_i?

Fisicks, I don't know what problem you have. The basic idea is simple: sort the q_i and s_i in order and put them into one big list. When a number comes from both lists, add the exponents; otherwise take the exponent from where it came (e_i or f_i). Example:

(3^2 * 2^1) * (7^2 * 2^2 * 3^3) =
(2^1 * 3^2) * (2^2 * 3^3 * 7^2) =
2^(1+2) * 3^(2+3) * 7^2 =
2^3 * 3^5 * 7^2
 
  • #4
greathouse why are you multiplying exponents, its just representing a and b to have the same bases of primes, and what i need help with is the proof part, not the what is the problem saying part.
 
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  • #5
Sorry, misread the problem. Your problem is easier -- just ignore the part about adding, and take missing primes' exponents to be 0.
 
  • #6
yeah i get that part! i need to know how you would write a formal proof to show that, cause that's what the problem is asking.
 
  • #7
Without knowing the requirements for your "formal" proof I couldn't possibly start. I don't imagine you mean what I mean when I say 'formal'.
 
  • #8
i don't know what more you want, the question is posted right up there in that link lol. all i want to know is how to answer that question.
 
  • #9
Fisicks said:
i don't know what more you want, the question is posted right up there in that link lol. all i want to know is how to answer that question.
I think you are having a problem of not being able to see a forest when a thousand trees are right in front of you. It is formal to state that a*b is a number that according to problem 3 has a unique prime fractorization, the rest is a fairly simple exercise.
 
  • #10
Fisicks said:
yeah i get that part! i need to know how you would write a formal proof to show that, cause that's what the problem is asking.

Yes, for a truly formal proof, see "Principia Mathematica". Not sure where the theorem in question is; probably in volume 2 or 3.
 
  • #11
Dragonfall said:
Yes, for a truly formal proof, see "Principia Mathematica". Not sure where the theorem in question is; probably in volume 2 or 3.
Dragonfall isn't it true that the only theorem needed is that provided by problem 3. I believe that problem 3 should be easy to give a formal proof. All the trees in problem 3 are those provided by "a" and "b" and how to obtain the forest that comprises them is simply to multiply "a" and "b". So what does problem three say about the product of "a" and "b"?
 
  • #12
Problem 3 isn't enough, it just "collects" equal primes into single powers. In problem 4 you have to "split" the product by introducing powers of 0. A proof along the lines suggested by CRGreathouse is what the problem's asking for, I think.

You can use problem 3 to say that a*b has a unique factorization, but you won't be able to say that a can be expressed with the prime factors of b directly, which is required for problem 3 to apply.
 
  • #13
Dragonfall said:
Problem 3 isn't enough, it just "collects" equal primes into single powers. In problem 4 you have to "split" the product by introducing powers of 0. A proof along the lines suggested by CRGreathouse is what the problem's asking for, I think.

You can use problem 3 to say that a*b has a unique factorization, but you won't be able to say that a can be expressed with the prime factors of b directly, which is required for problem 3 to apply.

I think all problem 4 was asking was to apply the fact that p^0 is equal to 1 to expand on the proof of problem 3. Call it an axiom or a proven fact or what ever, but I don't think problem 4 requires one to prove that p^0 is 1.
 
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  • #14
Not having seen the rest of the book, I don't know how problem 3 really relates to problem 4. But the fact is that to use the result of 3, you need to have the prime factorization. However, 1 isn't a prime, so you can't just use the fact that p^0=1.
 
  • #15
Dragonfall said:
Not having seen the rest of the book, I don't know how problem 3 really relates to problem 4. But the fact is that to use the result of 3, you need to have the prime factorization. However, 1 isn't a prime, so you can't just use the fact that p^0=1.

You don't understand because problem 4 specifically allows zero as the exponent of some of the primes in the prime factorization of a*b to be applied in the expression of "a" and "b" as an expression of common primes to a non-negative power multiplied together respectively. if p^6 is part of the prime factorization of a*b then the p part of "a" is p^i and that of "b" must be p^(6-i) where the i can go from 0 to 6.
 

What is the Fundamental Theorem of Arithmetic Problem?

The Fundamental Theorem of Arithmetic is a mathematical principle stating that every positive integer greater than 1 can be expressed as a unique product of prime numbers.

What is the significance of the Fundamental Theorem of Arithmetic Problem?

This theorem is significant because it shows that every positive integer has a unique prime factorization, which is crucial in many areas of mathematics, including number theory and cryptography.

How do you use the Fundamental Theorem of Arithmetic to find the prime factorization of a number?

To find the prime factorization of a number using the Fundamental Theorem of Arithmetic, you need to repeatedly divide the number by the smallest prime number that divides it evenly until the result is 1. The prime factors of the number will be all the prime numbers used in the division process.

What happens if a number is not a positive integer or is equal to 1?

If a number is not a positive integer or is equal to 1, the Fundamental Theorem of Arithmetic does not apply. This theorem only applies to positive integers greater than 1.

Are there any exceptions to the Fundamental Theorem of Arithmetic?

No, the Fundamental Theorem of Arithmetic holds true for all positive integers greater than 1. There are no exceptions to this theorem.

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