All possible ways to sum to a number

  • Thread starter Thread starter Aero51
  • Start date Start date
  • Tags Tags
    Sum
AI Thread Summary
The discussion explores methods to find all possible sums that equal a given number, highlighting the use of partitions in number theory. Rademacher's formula and Euler's theorem are mentioned as significant concepts, illustrating that the number of ways to express a number as a sum of distinct integers equals the number of ways to express it as a sum of odd integers. An iterative approach to sums is also discussed, noting that certain numbers, particularly powers of two, do not allow for linear iterative sums. Additionally, the "balls-in-boxes" problem is introduced as a combinatorial method for assigning sums, emphasizing the mathematical relationships involved. The conversation reflects a deep interest in the mathematical principles underlying summation and partitioning.
Aero51
Messages
545
Reaction score
10
I am curious if there is a universal formula to find all possible sums of a given number. For instance, to add to 10:
1+9
2+8
1+1+8
2+2+2+3+1, etc

I came up with a simple algorithm, but I'm sure there is something similar to Gauss's formula which can be utilized. I have heard Partitions used in number theory, but I don't know much about them beyond the fact that they are related to a similar problem.
 
Mathematics news on Phys.org
Yes, Rademacher's formula. See micromass's link.
 
There's a lovely theorem by Euler about the sums of a given number:

Count the number of ways a number may be represented as a sum of distinct integers. Then count the number of ways that number may be represented as a sum of odd but not necessarily distinct integers. The number of ways is the same in both cases.

For example take the number 7. Using distinct integers 7 may be represented as

7=7
=6+1
=5+2
=4+3
=4+2+1

5 ways.

Now we try with only the odd integers removing the restriction that they be distinct

7=7
=5+1+1
=3+3+1
=3+1+1+1+1
=1+1+1+1+1+1+1

5 ways. Unfortunately I can't remember the proof and I'm too lazy to have a bash at it myself.
 
Well here's something that might help (or not)

I did something with interative sums (liek 4 + 5 + 6, or 12+13, or 23+24+25+26+27) that always differ by one (and only integers)

So for 23, the only iterative sum is 11 + 12

For 26, its 5+6+7+8

Its basically related to the factors of the number...

Take 23 for instance... Its only factor is 1

Add 1 to the factor to get 2, then 23/2 = 11.5, then +- (1/2) from 11.5, to get 11 and 12

11 + 12 = 23

Take 35... the factors are 1,5,7,35...

So you can have a linear sum of 5 terms that revolve around 7 (5 + 6 + 7 + 8 + 9) or a linear sum of 7 terms that revolve around 5 (2+3+4+5+6+7+8)...


I can't remember the whole thing... but I remember that it was inpossible for a linear iterative sum for anything of 2^n, like 2,4,8,16...

I can't remember, I worked on it, but I can't remember what the exact thing I did...


Idk, just thought I'd mention it, since your question reminded me of it...



... or take
 
This is also called the problem of balls-in-boxes , i.e., you have n balls that you want to

put in k boxes. Line up the balls , together with the boxes. Every line-up corresponds

to an assignment of balls in boxes , e.g., if n=k , the line up : ball, box, ball, box,...

ball, box corresponds to the assignment of exactly one ball for each box. In total,

you have n+k-1 objects to assign ( n balls, k boxes, but subtract one, since

after k-1 boxes have been used, there is only one way of filling the k-th box)

any choice of k-1 spaces ( for the balls, or, by symmetry, of n spaces

where the boxes boxen? * will go ) is an assignment of balls to boxes.

There are (n+k-1) C (k-1) or (n+k-1) C n ways of doing this assignment.

If you want to guarantee that no boxes are empty, throw-in one ball on each

box and do the same process: you are left with : n+k-1-k = n-1 balls to put in

the same k-1 boxes in (n-1) C (k-1) ways.
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...

Similar threads

Replies
3
Views
4K
Replies
2
Views
2K
Replies
7
Views
3K
Replies
24
Views
3K
Replies
13
Views
4K
Back
Top