query: Integer part of a positive real number exists and is unique
positive_passages.docid: 17285
positive_passages.text: Proof of greatest integer theorem: floor function is well-defined
negative_passages.docid: 22154
negative_passages.text: Show that given a set of positive n integers, there exists a non-empty subset whose sum is divisible by n
negative_passages.docid: 18108
negative_passages.text: Show if $(a,p)=1$ there is a unique inverse of $a$ modulo $p$
negative_passages.docid: 31900
negative_passages.text: Induction on Real Numbers
negative_passages.docid: 39698
negative_passages.text: On uniqueness of sums of prime powers
negative_passages.docid: 100565
negative_passages.text: Property of sup of a set of numbers
negative_passages.docid: 143067
negative_passages.text: Show given any $x\in\mathbb{R}$ show there exists a unique $n\in\mathbb{Z}$ such that $n-1\leq x <n$.
negative_passages.docid: 98002
negative_passages.text: Let $x$ be a positive real number. Inequality problem with $\pi$ and $x$ terms.
negative_passages.docid: 56528
negative_passages.text: On the proof of existence and uniqueness of radix representation
negative_passages.docid: 152172
negative_passages.text: Does there exist a positive irrational number $\alpha $, such that for any positive integer $n$ the number $\lfloor n\alpha \rfloor$ is not a prime?
negative_passages.docid: 107990
negative_passages.text: Does there exist an integer $x$ satisfying the following congruences?
negative_passages.docid: 114245
negative_passages.text: What might I use to show that an entire function with positive real parts is constant?
negative_passages.docid: 135955
negative_passages.text: Prove: The positive integers cannot be partitioned into arithmetic sequences (using Complex Analysis)
negative_passages.docid: 135750
negative_passages.text: Prove that if a and b are integers, then there are unique integers q and r such that $a = bq + r$, $-|b|/2 < r \le |b|/2$
negative_passages.docid: 105405
negative_passages.text: Product of three consecutive positive integers is never a perfect power
negative_passages.docid: 134352
negative_passages.text: The number of possible factorizations of a positive integer.
negative_passages.docid: 142000
negative_passages.text: Prove that there is an irrational number and a rational number between any two distinct real numbers