query: Number of integer solutions to $a_1\ge a_2\ge\ldots\ge a_i\gt 0$ such that $a_1+a_2+\ldots+a_i=n$
positive_passages.docid: 156542
positive_passages.text: Number of solutions to $a_1 + a_2 + \dots + a_k = n$ where $n \gt 0$ and $0 \lt a_1 \leq a_2 \leq \dots \leq a_k$ are integers.
negative_passages.docid: 7466
negative_passages.text: How would I prove that if an integer $n>2$, then $\bar{z}=z^{n-1}$ has $n+1$ solutions
negative_passages.docid: 123159
negative_passages.text: Solutions of the diophantine equation $x+y+z=xyz$?
negative_passages.docid: 59479
negative_passages.text: For a positive integer $n\geq 2$ with divisors $1=d_1<d_2<\cdots<d_k=n$, prove that $d_1d_2+d_2d_3+\cdots+d_{k-1}d_k<n^2$
negative_passages.docid: 24837
negative_passages.text: $x^2+1=0$ uncountable many solutions
negative_passages.docid: 76696
negative_passages.text: Using induction to prove $2^{n-1}(1 + a_1a_2\ldots a_n) \geq (1+a_1)(1+a_2)\ldots(1+a_n)$ for $a_i \geq 1$
negative_passages.docid: 128107
negative_passages.text: find all $ a \in \mathbb N $ such that $2^a − 1 \equiv 0 \pmod 3$ and $ 4x^2 + 1 \equiv 0 \pmod {\frac{2^a -1}{3}}$ has a solution
negative_passages.docid: 82571
negative_passages.text: Number of solutions of $x^2=1$ in $\mathbb{Z}/n\mathbb{Z}$
negative_passages.docid: 112575
negative_passages.text: Finding integer solutions of $m^2-n^5 = m - n$
negative_passages.docid: 115244
negative_passages.text: Integer solutions of $m^2+(m+1)^2=n^4+(n+1)^4$
negative_passages.docid: 95494
negative_passages.text: Number of solutions of an arithmetic function's equation
negative_passages.docid: 105302
negative_passages.text: Solutions of bivariate cubic Diophantine equation
negative_passages.docid: 1001
negative_passages.text: Number of integer solutions of $x^2 + y^2 = k$
negative_passages.docid: 106271
negative_passages.text: Solutions of Diophantine equations in Natural numbers
negative_passages.docid: 146491
negative_passages.text: Number of solutions of $x_1+x_2+\dots+x_k=n$ with $x_i\le r$
negative_passages.docid: 59091
negative_passages.text: A quartic diophantine equation
negative_passages.docid: 114379
negative_passages.text: Show that number of solutions satisfying $x^5=e$ is a multiple of 4?