Integer Representation Through Multiplication of Integers

[Jim Phillips]

Hello,

Can an integer always be represented through the multiplication of two or more integers? (Are all integers divisible by some set of 2 or more integers (- or +)?)

For example, 8 is can be represented by 1 x 8, 2 x 4 and 2 x 2 x 2. But what about 257 or even - integers?

I'm trying to determine:
1) Can an integer always be represented through the multiplication of two or more integers.

2) If they can - Starting with a given integer, what would be the formula to determine the least amount of integers that multiple up to its value? (e.g. for 8 it's 1 x 8 and 2 x 4 - but what is a formula that could help me determine this?)

TIA
-JP

 

[Gokul43201]

1) Any integer z can be written as z=1*z, so yes.
2) Since the set {1,z} has 2 elements, the least number is always 2.

Is this what you're really looking for?

 

[Jim Phillips]

Good point, I should have been more clear.

Sorry about that.

1 x 257, is correct, but I'm looking for the other sets of possible numbers as well. I'm interested in studying many integers and the sets of integers that multiple up to them.

So, how can I determine the other sets of integers that multiply up to, for example, 257?

Thanks again
-JP

 

[Jim Phillips]

Factors may help me ... I"m not sure

For example, the factors of 57 are: -57, -19, -3, -1, 1, 3, 19, and 57.

So I can get 57 a few ways including 1 x 3 x 19.

I was hoping there we a few more sets that would work for numbers like 257, but it seems that only integers that multiple up to 257 are -257, -1, 1, and 257 so I may be stuck with 1 x 257 with no possibility of a set of three or more numbers (e.g. Z x Z x Z) multiplying up to 257.

 

[mathman]

I think you should try to understand what prime numbers are and the concept of unique factorization.

 

[CRGreathouse]

I was hoping there we a few more sets that would work for numbers like 257, but it seems that only integers that multiple up to 257 are -257, -1, 1, and 257 so I may be stuck with 1 x 257 with no possibility of a set of three or more numbers (e.g. Z x Z x Z) multiplying up to 257.

257 is prime, so all factorizations will be:
1. 257 * an even number of -1s * any number of 1s
2. -257 * an odd number of -1s * any number of 1s

 

[Jim Phillips]

Thank you for the feedback CRGreathouse, mathman and Gokul43201.
-JP

 

[[Quadratic]]

Any prime number p can be represented as the product of 1 and p.

All other integers can be represented as the product of primes.

 

[HallsofIvy]

So, if you are not accepting 1*n as "a product of two or more integers", then
1) If n is prime, it cannot be written as a product except as 1*p
2) If n is composite, it can be.
3) 1, which is neither prime nor composite cannot be written as a product of two integers except as 1*1.
That is basically the definition of "prime" and "composite".

 

[Take_it_Easy]

well, in my modest opinion, since
1 = (-1)(-1) = (-1)(-1)1 = (-1)(-1)(-1)(-1) = ... = ((-1)^(2n))(1^m)
for every n,m natural,

you should put your question under some rule of irredundance to try to find a interesting answer.

 

[ramsey2879]

well, in my modest opinion, since
1 = (-1)(-1) = (-1)(-1)1 = (-1)(-1)(-1)(-1) = ... = ((-1)^(2n))(1^m)
for every n,m natural,

you should put your question under some rule of irredundance to try to find a interesting answer.

I am all for formulating rules of irredundance, For instance Greathouse omits the negative numbers since negative numbers can be obtain by adding an odd number the factors -1either alone or combined with positve factors of the number.
Since 1 can be a factor an infinite times another rule would be that 1 can be a factor at most only once.
Another rule would be to treat a*b and b*a as being indistinct. That is the order of the factors would be regarded as immaterial as to distinctness.
Still the above rules do not preclude the redundance of having some positive integers with the property of having more than one way to be factored. For instance 8 can be factored as 1*8 or as 2*4. This is where the definition of prime integers comes into play. One property of prime numbers n is that prime integers, under the preceeding rules, can only be factored into more than one factor as 1*n. But there is a issue here if we regard 1 as being a prime number since we still can not postulate the number one rule of number theory under the preceding rules. That is we still can not say that every positive integer can be factored as a product prime numbers in one and only way. For instance 6 can be written as 1*2*3 or simply as 2*3. ( 2 and 3 are clearly prime numbers under the preceding rules since
each can be written as 1*3 or 1*5 but 1*1 violates the rule that "1" can be factor at most only once. There is a way to address this issue. 1) Make the exception for the rule re using one as a factor twice only for the product 1*1, and regard all products as containing the factor "prime factor" 1 at least once, (1 = 1*1, 2 = 1*2, 3 = 1*3, ... 6 = 1*2*3 etc). That rule however would result in the redundancy of having to use 1 as a factor for every product, both primes and composite numbers. Therefore, the convention is to exclude regarding 1 as a "prime" and to regard prime numbers 2,3,5 etc. as products in themselves.
Thus the represention of integers above 1 as a product of prime factors is (2),(3),(2*2),(5), (2*3),(7),(2*2*2) etc.