What about the composites?

February 13th, 2010 | by admin |

Primes are numbers with two factors and there’s no formula for the nth one, right?
Well, what about other numbers with 4, 6, 8… (I exclude odd numbers as these are square numbers and therefore very rare) factors can they predict the nth one of these? I think not. Why are these not considered or even named. I know primes how primes are involved in the fundamental theorem of arithmetic and everything but have not been able to find a single piece of research concerning the number of divisors had by other numbers. Please explain or point me in the direction of research.
I don’t mean nth composite, I mean say if numbers with 4 factors were called quarmes then could we predict where the nth quarme would lie more easily than the nth prime.

The nth composite is very easy to determine. It is at least 1 + the nth-1 composite, and at most two. The composites are not the subject of much investigation, for their pattern is obvious and unlike the primes, their distribution does not give insight into the overall pattern of the integers. Furthermore it is unknown if there is a formula for predicting the nth prime, it is still under investigation.

  1. 5 Responses to “What about the composites?”

  2. By arty.fishal on Feb 14, 2010 | Reply

    The nth composite is very easy to determine. It is at least 1 + the nth-1 composite, and at most two. The composites are not the subject of much investigation, for their pattern is obvious and unlike the primes, their distribution does not give insight into the overall pattern of the integers. Furthermore it is unknown if there is a formula for predicting the nth prime, it is still under investigation.
    References :

  3. By brilliantwaz on Feb 14, 2010 | Reply

    let us take 8.
    8=2^3
    number of factors of 8 are 1,2,4,8 which are 4.
    we can observe that it is 3+1=4.
    take 12
    12=2^2*3
    factors of 12 are 1,2,3,4,6,12 which are 6.
    we can observe that it has (2+1)(1+1)=6 divisors.
    there is a theorem in number theory that a composite number greater then 2,can be written as product of prime numbers . suppose a number can be written as a^x*b^y*c^z, where a,b,c are prime factors of the number and x,y,z are their powers, then the number of divisors or factors of the number are (x+1)(y+1)(z+1).the above examples may clear this point and may help u in ur research.
    References :

  4. By thefreevariable on Feb 14, 2010 | Reply

    Prime numbers have lots of interesting properties in algebra and number theory. If I want to study the properties of a composite number, the first thing I want to know is its prime factorization. That’s why composite numbers receive so little attention; they are treated as the product of their prime factors, and that takes us full-circle back to studying primes.

    As for the number of divisors for a given composite n, I suppose you could write n in terms of its prime factors and then make consider all possible products of subsets of those factors. That would generate all possible factors.

    You might also want to take a look at Euler’s phi-function (sometimes called totient function). I’ve provided a link below. The function, applied to a composite n, gives you the number of integers less than n that share no divisors with n (except 1). Conversely, the cototient of n is the number of integers less than n that share at least one divisor with n.

    EDIT: I just read your additional details. That’s a very interesting question in my opinion. I don’t know the answer, but if you’re really into number theory, perhaps you could solve it and enlighten us!
    References :
    http://en.wikipedia.org/wiki/Euler_phi_function

  5. By knashha on Feb 14, 2010 | Reply

    A great question. The composites per se are virtually ignored.
    The composites have a genesis which is pretty obvious:
    multiply all the numbers in (2,3,4,5,…..) by all the numbers in (2,3,4,5,…..) and each composite will appear but no primes.
    Because of this, the value of the nth composite, A(n)
    grows slightly faster than n and the space created is filled
    with primes. From this point of view it would seem easier to find the value of the nth composite than to find the value of the nth prime. It is difficult to order the results of the product

    (2,3,4,5,…..)(2,3,4,5,…..)

    and predict the size of the nth term. If it were possible then
    we could find the number of primes not exceeding n to be

    A(n) – n – 1 .

    This line would be an attempt to understand the growth of the primes without refering to smaller primes but anchoring the
    investigation in the global genesis of the composite numbers.
    References :

  6. By bh8153 on Feb 14, 2010 | Reply

    You are wondering about something related to the omega function, as described in the web page below.

    I can see no evidence on this page of investigations of the type you suggest, as to the density distribution and average separation of those numbers which all have the same value of omega, ignoring the numbers with a different value.
    References :
    http://mathworld.wolfram.com/DistinctPrimeFactors.html

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