query: Hardy Ramanujan Question
positive_passages.docid: 147499
positive_passages.text: Mathematica Function Taxicab Numbers
negative_passages.docid: 14724
negative_passages.text: For primes $p≡3\pmod 4$, prove that $[(p−1)/2]!≡±1\pmod p$.
negative_passages.docid: 10872
negative_passages.text: Solving Hardy Weinberg problems
negative_passages.docid: 52427
negative_passages.text: Prove, $\sqrt{n} \le (n!)^\frac{1}{n} \le \frac{n+1}{2} $
negative_passages.docid: 31485
negative_passages.text: Three negative and three positive $1$s in a serie
negative_passages.docid: 126202
negative_passages.text: Analysis Question Involving Real Numbers
negative_passages.docid: 150101
negative_passages.text: Riemann Sum of $\sqrt{x}$ with uniform partition
negative_passages.docid: 38876
negative_passages.text: $r! \equiv (−1)^k \pmod p$
negative_passages.docid: 19804
negative_passages.text: How to prove that $|w_{n+1}-w_n| \leq (2/3)^2 |w_n - w_{n-1}|$?
negative_passages.docid: 127291
negative_passages.text: Show that $\left( \frac{q}{p} \right) \equiv q^{(p-1) / 2} \mod p$, where $\left( \frac{q}{p} \right)$ is the Legendre Symbol
negative_passages.docid: 80826
negative_passages.text: To prove $\left|\frac{p_n(z)}{q_m(z)}\right|\leq \frac{M}{|z|^{m-n}}$ for some $M>0$
negative_passages.docid: 107258
negative_passages.text: Proving If $k \le \left\lfloor \frac{n}{2} \right\rfloor$ then $\binom{n}{k-1} < \binom{n}{k}$
negative_passages.docid: 155190
negative_passages.text: Stone-Weierstrass theorem with $p(1/x)$
negative_passages.docid: 100566
negative_passages.text: Ramanujan's approximation to factorial
negative_passages.docid: 114404
negative_passages.text: For which positive integer values of $n$ does $a|mn-b?$
negative_passages.docid: 16697
negative_passages.text: Fermat Last Theorem for non Integer Exponents
negative_passages.docid: 146546
negative_passages.text: Proving for all integer $n \ge 2$, $\sqrt n < \frac{1}{\sqrt 1} + \frac{1}{\sqrt 2}+\frac{1}{\sqrt 3}+\cdots+\frac{1}{\sqrt n}$