query: Number of solutions for $x_1+ x_2+ x_3 + \cdots + x_k=n$, where $0\leq x_i\leq p$ for all $i$
positive_passages.docid: 146491
positive_passages.text: Number of solutions of $x_1+x_2+\dots+x_k=n$ with $x_i\le r$
negative_passages.docid: 127945
negative_passages.text: Number of solutions to the congruence $x^q \equiv 1 \mod p$.
negative_passages.docid: 142360
negative_passages.text: How to show $x_1,x_2, \dots ,x_n \geq 0 $ and $ x_1 + x_2 + \dots + x_n \leq \frac{1}{2} \implies (1-x_1)(1-x_2) \cdots (1-x_n) \geq \frac{1}{2}$
negative_passages.docid: 87479
negative_passages.text: Closed form for $\sum_{k=0}^{n-p}\binom{n}{k}\binom{n}{p+k}$
negative_passages.docid: 112390
negative_passages.text: Solutions of $x^2 \equiv 1 \pmod N$
negative_passages.docid: 39993
negative_passages.text: Count the number of positive solutions for a linear diophantine equation
negative_passages.docid: 103165
negative_passages.text: the number of the positive integer numbers $k$ that makes for the two quadratic equations $ \pm x^2 \pm px \pm k$ rational roots.
negative_passages.docid: 94700
negative_passages.text: Integer solutions for $x^3+2=y^2$?
negative_passages.docid: 16035
negative_passages.text: Irreducibility of $X^{p-1} + \cdots + X+1$
negative_passages.docid: 33314
negative_passages.text: How many non-negative integer solutions does $x_1+x_2+\cdots+x_n=A$ have?
negative_passages.docid: 147596
negative_passages.text: Number of solutions to $x^n=e$ in group $G$ is divisible by $n$
negative_passages.docid: 44354
negative_passages.text: Total number of solutions of an equation
negative_passages.docid: 82137
negative_passages.text: Given a, b How many solutions exists for x, such that: $a \bmod{x}=b $
negative_passages.docid: 151732
negative_passages.text: Number of solutions of polynomials in a field
negative_passages.docid: 130565
negative_passages.text: Find all solutions to $x^3+(x+1)^3+ \dots + (x+15)^3=y^3$
negative_passages.docid: 38541
negative_passages.text: How many solutions are there to $x_1 + x_2 + ... + x_5 = 21$?
negative_passages.docid: 16946
negative_passages.text: How many solutions for an equation with simple restrictions