query: Proof that the sum of the first $n$ cubes is equal to the square of the $n^{th}$ triangular number
positive_passages.docid: 93657
positive_passages.text: Proving “The sum of $n$ consecutive cubes is equal to the square of the sum of the first $n$ numbers.”
negative_passages.docid: 135458
negative_passages.text: When is a sum of consecutive squares equal to a square?
negative_passages.docid: 54900
negative_passages.text: How to prove $|\sum_{i=1}^n a_i|\le \sqrt{n} \sqrt{\sum_{i=1}^n a_i^2}$
negative_passages.docid: 94389
negative_passages.text: Proving formula for product of first n odd numbers
negative_passages.docid: 65965
negative_passages.text: Theoretical way to prove no positive integer $n$ exists such that $n+3$ and $n^2+3n+3$ are both perfect cubes.
negative_passages.docid: 80671
negative_passages.text: Product of sums of square is a sum of squares.
negative_passages.docid: 22840
negative_passages.text: The sum of the first $n$ squares is a square: a system of two Pell-type-equations
negative_passages.docid: 90076
negative_passages.text: Prove that every positive integer $n$ is a unique product of a square and a squarefree number
negative_passages.docid: 92106
negative_passages.text: Prove that $n$ is a sum of two squares?
negative_passages.docid: 17714
negative_passages.text: Proof $\sum_{k=1}^n (-1)^{k+1} \binom{n}{k}\frac{1}{k} = H_n$ by induction
negative_passages.docid: 69131
negative_passages.text: Different proofs of $\,a^n-b^n =(a-b)\sum_{i=0}^{n-1} a^i b^{n-1-i} $?
negative_passages.docid: 10920
negative_passages.text: $\delta_2(n)$ the sum of the squares of the positive divisors of $n$.
negative_passages.docid: 37569
negative_passages.text: Prove that $\binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{r+1} $
negative_passages.docid: 143113
negative_passages.text: Non-inductive, not combinatorial proof of $\sum_{i \mathop = 0}^n \binom n i^2 = \binom {2 n} n$
negative_passages.docid: 146847
negative_passages.text: Given a square matrix A of order n, prove $\operatorname{rank}(A^n) = \operatorname{rank}(A^{n+1})$
negative_passages.docid: 5692
negative_passages.text: Proving that there is a perfect square between $n$ and $2n$
negative_passages.docid: 92440
negative_passages.text: Let $n \in \mathbb{Z}^+$, prove the identity $ \sum_{k=1}^{n-1} \binom {n} {k} \frac{kn^{n-k}}{k+1}=\frac{n(n^{n}-1)}{n+1}$