All Questions
Tagged with integer-partitions diophantine-equations
15
questions
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"Factorization" of the solutions set of a system of linear diophantine equations over non-negative integers
Suppose we have a system of linear diophantine equations over non-negative integers:
$$
\left\lbrace\begin{aligned}
&Ax=b\\
&x\in \mathbb{Z}^n_{\geq0}
\end{aligned}\right.
$$
where $A$ is a ...
- 11
1
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1
answer
288
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A certain conjectured criterion for restricted partitions
Given the number of partitions of $n$ into distinct parts $q(n)$, with the following generating function
$\displaystyle\prod_{m=1}^\infty (1+x^m) = \sum_{n=0}^\infty q(n)\,x^n\tag{1a}$
Which may be ...
- 2,813
2
votes
3
answers
211
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Is there a way to find the values of 3 variables with just one equation?
I was doing a personal project until I came upon this equation that i need to solve to continue, the thing is I don’t thing I have been thought this in math ever so i wonder if this even possible to ...
- 21
1
vote
1
answer
122
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Number of solutions number theory problem
I am wondering how many nonnegative solutions the following Diophantine equation has: $$x_1+x_2+x_3+\dots+x_n=r$$ if $x_1 \leq x_2 \leq x_3 \leq \dots \leq x_n$
I know if a sequence can be non-...
1
vote
2
answers
321
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Number of non-negative integer solutions of linear Diophantine equation
Given $n \ge 0$, find the number of non-negative integer solutions of the following equation $$2x_1+x_2+x_3+x_4+x_5=n$$
I tried putting one $x_1$ on the right-hand side, as follows
$$x_1+x_2+x_3+x_4+...
- 533
6
votes
0
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198
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The smallest non-zero integer $c$ such that $\sum\limits_{n=1}^m\pm(x+n)^6 = c$?
We have the neat equalities,
I. Group 1
For $k=2,3,4,5,\dots$
$$\sum_{n=1}^{2^k}\epsilon_n(x+n)^k = 2^{\frac{k(k-1)}{2}}k! = 4,\;48,\;1536,\;\color{brown}{122880},\dots$$
for appropriate $\...
- 54.9k
5
votes
1
answer
567
views
Partition problem for consecutive $k$th powers with equal sums (another family)
This is the partition problem as applied to a special set, namely the first $n$ $k$th powers. Assume the notation,
$$[a_1,a_2,\dots,a_n]^k = a_1^k+a_2^k+\dots+a_n^k$$
I. Family 1
The following ...
- 54.9k
1
vote
3
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423
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Sums of Consecutive Cubes (Trouble Interpreting Question)
Show that it is possible to divide the set of the first twelve cubes $\left(1^3,2^3,\ldots,12^3\right)$ into two sets of size six with equal sums.
Any suggestions on what techniques should be used to ...
- 2,088
33
votes
5
answers
57k
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Counting bounded integer solutions to $\sum_ia_ix_i\leqq n$
I want to find the number of nonnegative integer solutions to
$$x_1+x_2+x_3+x_4=22$$
which is also the number of combinations with replacement of $22$ items in $4$ types.
How do I apply stars and bars ...
1
vote
2
answers
731
views
Integer solutions
How many positive integer solutions are there to $x_1 + x_2 + x_3 + x_4 < 100$?
I haven't seen any problems with "less than", so I'm a bit thrown off. I'm not sure if my answer is correct, but if ...
- 3,224
3
votes
2
answers
254
views
Count the number of integer solution to $\sum_{i=i}^{n}{f_ig_i} \geq 5 $
How to count the number of integer solutions to $$\sum_{i=i}^{n}{f_ig_i} \geq 5$$ such that $\displaystyle \sum_{i=1}^{n}{f_i}=6$ , $\displaystyle \sum_{i=1}^{n}{g_i}=5$ , $\displaystyle 0 \leq f_i \...
- 247
6
votes
1
answer
234
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When is the sum of first $n$ numbers equal to the sum of the next $k$ numbers?
When is the sum $1+2+\cdots + n = (n+1) + (n+2) + \cdots +(n+k)$?
The easiest solution $(n,k)$ is $(2,1)$. For example, $1+2 = 3$. Do any others exist?
Roots of $(n+k)^2 + (n+k) = 2n^2 +2n$ give ...
- 63
1
vote
1
answer
313
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Total number of solutions of an equation
What is the total number of solutions of an equation of the form $x_1 + x_2 + \cdots + x_r = m$ such that $1 \le x_1 < x_2 < \cdots < x_r < N$ where $N$ is some natural number and $x_1, ...
- 11
15
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5
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Count the number of integer solutions to $x_1+x_2+\cdots+x_5=36$ [duplicate]
How to count the number of integer solutions to $x_1+x_2+\cdots+x_5=36$ such that $x_1\ge 4,x_3 = 11,x_4\ge 7$
And how about $x_1\ge 4, x_3=11,x_4\ge 7,x_5\le 5$
In both cases, $x_1,x_2,x_3,x_4,...
- 2,060
2
votes
1
answer
916
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Number of solutions of Frobenius equation
I have one problem which needs to count the number of solution of the equation $$2x+7y+11z=42$$
where $x,y,z \in \{0,1,2,3,4,5,\dots\}$.
My attempt:
I noticed that that maximum value of $z$ could ...
- 22.5k