Partitioning with Primes: Comparing P(n) and P'(n)

In summary: So for the two cases:(1) 1 x 1 + 2 x 1 + 3 x 1 + 4 x 1...= V, and(2) 1 x 2 + 2 x 3 + 3 x 5 + 4 x 7...= V (where the numbers are primes).In summary, the conversation is discussing the number of ways to write a given number as the sum of positive integers, with a comparison between the number of ways for conventional partitions and for partitions using only prime numbers. The conversation also touches on the difficulty of determining the "initial exact packing" for boxes filled with blocks of either sequential integer or prime lengths.
  • #1
Loren Booda
3,125
4
If a partition P(n) gives the number of ways of writing the integer n as a sum of positive integers, comparatively how many ways does the partition P'(n) give for writing n as a sum of primes?
 
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  • #2
doesnt it varies from number to number for example the partition of the number 10 by the sums of prime numbers is 5+5,2+3+5,3+7,2+2+2+2+2 so P'(10)=4 (if mistaken do correct me) and the number of partitions of let's say 15 by its prime numbers sums is diifferent from those of 10.
 
  • #3
loop quantum gravity,

Yes, I believe the number of "prime partitions," P'(n), increases with integer n, just not as rapidly as that of conventional partitions, P(n). (Do I understand you correctly?)
 
  • #4
The number of ways that a number can be written as the sum of positive integers? I assume that you mean without ordering.

So we have:
N(0)=0
N(1)=1 (1)
N(2)=2 (1+1,2)
N(3)=3 (1+1+1,1+2,3)
N(4)=5 (1+1+1+1,1+1+2,1+3,2+2,4)
N(5)=7 (1+1+1+1+1,1+1+1+2,1+1+3,1+2+2,1+4,2+3,5)
N(6)=10(1+1+1+1+1+1,1+1+1+1+2,1+1+1+3,1+1+2+2,1+1+4
1+2+3,1+5,2+2+2,2+4,6)

P(0)=0
P(1)=0
P(2)=1 (2)
P(3)=1 (3)
P(4)=1 (2+2)
P(5)=2 (2+3,5)
P(6)=2 (2+2+2,3+3)
P(7)=3 (2+2+3,2+5,7)
P(8)=3 (2+2+2+2,2+3+3,3+5)
P(9)=4 (2+2+2+3,2+2+5,2+7,3+3+3)

Obviously P(n)<N(n) and
[tex]\lim_{n \rightarrow \infty} \frac{P(n)}{N(n)}=0[/tex]
 
  • #5
but one itself isn't a prime.
 
  • #6
Take a box of volume V, exactly filled by a large number of either (1.) blocks having progressively integer length, or (2.) blocks having progressively prime length and both (1. & 2.) of unit square cross-section. Is the initial exact packing more easily determined for one situation than the other?
 
  • #7
Originally posted by Loren Booda
Take a box of volume V, exactly filled by a large number of either (1.) blocks having progressively integer length, or (2.) blocks having progressively prime length and both (1. & 2.) of unit square cross-section. Is the initial exact packing more easily determined for one situation than the other?

You mean that you have a line segment, and you're partitioning it into intervals of decreasing size?

I don't understand the notion of 'initial exact packing' that you describe, but there are definitely more possibe arrangements for (1) than there are for (2) if V > 0.
 
  • #8
Two sets of blocks each fit a given box exactly. All blocks have a square cross-section of unit area. The first set comprises blocks of sequential integer >0 length, the second set comprises blocks of sequential prime >1 length. Initially given either set unboxed, which boxing is more easily determinable?
 
  • #9
Huh? I don't understand your question.

Are you trying to do this type of problem:

Given an integer N > 1 construct a set of primes [tex]{p_i}[/tex] with [tex]i \neq j \rightarrow p_i \neq p_j[/tex] and [tex]\sum p_i = N[/tex].
 
  • #10
Sorry, NateTG, I perceived a pattern that apparently wasn't there.
 

1. What is the significance of partitioning with primes?

Partitioning with primes is a mathematical concept that involves breaking down a number (n) into smaller parts using only prime numbers. This has been an important area of study in number theory and has applications in various fields such as cryptography and coding theory.

2. How is P(n) different from P'(n)?

P(n) and P'(n) are both functions that represent the number of ways a number (n) can be partitioned into smaller parts. However, P(n) includes all possible partitions, while P'(n) only considers partitions that use distinct numbers.

3. What is the relationship between P(n) and P'(n)?

The relationship between P(n) and P'(n) is that P(n) can be expressed as a sum of P'(n) for all values of n. In other words, P(n) = P'(n) + P'(n-1) + P'(n-2) + ... + P'(1).

4. How does the growth rate of P(n) compare to P'(n)?

The growth rate of P(n) is significantly larger than that of P'(n). This is because P(n) includes all possible partitions, while P'(n) only considers partitions with distinct numbers, which are fewer in number.

5. What are some real-world applications of partitioning with primes?

Partitioning with primes has various real-world applications, such as in cryptography where it is used to generate secure encryption keys. It is also used in coding theory to create efficient error-correcting codes. In addition, partitioning with primes has been studied in economics to understand the distribution of wealth among individuals.

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