query: Does there exist a positive integer $k$ such that $A = 2 ^ k + 3 ^ k$ is a square number?
positive_passages.docid: 145558
positive_passages.text: Prove that $2^n+3^n $ is never a perfect square
negative_passages.docid: 31436
negative_passages.text: Show that for all $a\in\mathbb{N}$, there exists $b\in\mathbb{N}$ and square-free integer $c$ such that $\sqrt{a}=b\sqrt{c}$.
negative_passages.docid: 513
negative_passages.text: If n is an odd integer, show there exists a positive integer k such that 2^k mod n = 1.
negative_passages.docid: 63383
negative_passages.text: Integers $n$ for which $|2^n + 5^n – 65|$ is a perfect square
negative_passages.docid: 122207
negative_passages.text: Use the Division Algorithm to show the square of any integer is in the form $3k$ or $3k+1$
negative_passages.docid: 98443
negative_passages.text: Find positive integers such that $2n^3 + 5|n^4 +n+1$
negative_passages.docid: 134121
negative_passages.text: Determining all the positive integers $n$ such that $n^4+n^3+n^2+n+1$ is a perfect square.
negative_passages.docid: 107990
negative_passages.text: Does there exist an integer $x$ satisfying the following congruences?
negative_passages.docid: 58171
negative_passages.text: Two problems about Squarefree numbers
negative_passages.docid: 135212
negative_passages.text: The square of an integer has form $3n$ or $3n+1$
negative_passages.docid: 28781
negative_passages.text: $k^{2}+(k+1)^{2}$ being a perfect square for infinitely many $k$
negative_passages.docid: 55787
negative_passages.text: Is there any matrix $2\times 2$ such that $A\neq I$ but $ A^3=I$?
negative_passages.docid: 26062
negative_passages.text: Given $a>b>2$ both positive integers, which of $a^b$ and $b^a$ is larger?
negative_passages.docid: 137967
negative_passages.text: Are there $a,b \in \mathbb{N}$ that ${(\sum_{k=1}^n k)}^a = \sum_{k=1}^n k^b $ beside $2,3$
negative_passages.docid: 71935
negative_passages.text: Let $k$ be a natural number . Then $3k+1$ , $4k+1$ and $6k+1$ cannot all be square numbers.
negative_passages.docid: 134765
negative_passages.text: Is an integer a sum of two rational squares iff it is a sum of two integer squares?
negative_passages.docid: 91113
negative_passages.text: $ \exists a, b \in \mathbb{Z} $ such that $ a^2 + b^2 = 5^k $