query: What is the correct way to write in first-order formula that for any natural number $n$, $P(n)$ is true?
positive_passages.docid: 28281
positive_passages.text: Different standards for writing down logical quantifiers in a formal way
negative_passages.docid: 145519
negative_passages.text: Prove that for any natural number $n>1$, we have $(2n)!<(n(n+1))^n$
negative_passages.docid: 89938
negative_passages.text: Is the Formula Logically Valid?
negative_passages.docid: 58264
negative_passages.text: How to derive the formula for the sum of the first $n$ perfect squares?
negative_passages.docid: 48051
negative_passages.text: Proof related to natural numbers
negative_passages.docid: 96341
negative_passages.text: Proving $v(s,p)=2^{p-1}(2s-1)$ is a bijection of natural numbers and $f(s)=2s-1$ is a bijection between natural numbers and odd numbers.
negative_passages.docid: 58079
negative_passages.text: How does $ \sum_{p<x} p^{-s} $ grow asymptotically for $ \text{Re}(s) < 1 $?
negative_passages.docid: 131385
negative_passages.text: How to provide a counterexample for if $n$ is prime, then $2^n -1$ is prime
negative_passages.docid: 4512
negative_passages.text: Is there a direct, elementary proof of $n = \sum_{k|n} \phi(k)$?
negative_passages.docid: 96846
negative_passages.text: Decimal representation begins with $N$
negative_passages.docid: 43044
negative_passages.text: All natural numbers are equal.
negative_passages.docid: 41354
negative_passages.text: Binomial-coefficients if, k, m, n natural numbers and k \leq n the result of
negative_passages.docid: 123427
negative_passages.text: Proving the sum of the first $n$ natural numbers by induction
negative_passages.docid: 129860
negative_passages.text: Find a formula for this sequence (and prove it).
negative_passages.docid: 94620
negative_passages.text: Which number was removed from the first $n$ naturals?
negative_passages.docid: 137669
negative_passages.text: $P$ is a monic polynomial of degree $n$ , then which are correct?
negative_passages.docid: 135351
negative_passages.text: $C(n,p)$: even or odd?