query: Prove that there is a unique positive integer $n$ such that $2^8 + 2^{11} + 2^n$ is a perfect square.
positive_passages.docid: 23523
positive_passages.text: Two problems on number theory
negative_passages.docid: 67968
negative_passages.text: When is $2^n -7$ a perfect square?
negative_passages.docid: 9146
negative_passages.text: Proving that a list of perfect square numbers is complete
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: 63383
negative_passages.text: Integers $n$ for which $|2^n + 5^n – 65|$ is a perfect square
negative_passages.docid: 50927
negative_passages.text: If $m,n\in N$ Prove that there is such a positive integer k, such that $(\sqrt{m}+\sqrt{m+1})^n=\sqrt{k}+\sqrt{k+1}$
negative_passages.docid: 156272
negative_passages.text: Unique pair of positive integers $(p,n)$ satisfying $p^3-p=n^7-n^3$ where $p$ is prime
negative_passages.docid: 99606
negative_passages.text: Prove or disprove there exists a set of four numbers out of the $48$ positive integers of which product is a perfect square
negative_passages.docid: 60738
negative_passages.text: If 4n+1 and 3n+1 are both perfect sqares, then 56|n. How can I prove this?
negative_passages.docid: 55206
negative_passages.text: Prove that the equation $x^4+(2k^2)x^2+2k^4$ is not a perfect square
negative_passages.docid: 142580
negative_passages.text: Prove that a perfect square is either a multiple of $4$ or of the form $4q+1$.
negative_passages.docid: 147666
negative_passages.text: How to prove that either $2^{500} + 15$ or $2^{500} + 16$ isn't a perfect square?
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: 72511
negative_passages.text: If $n$ is an odd natural number, then $8$ divides $n^{2}-1$
negative_passages.docid: 24954
negative_passages.text: Prove that there are infinitely many integers of the form $111\ldots111$ that are multiple of $n$.
negative_passages.docid: 63182
negative_passages.text: For every $n \ge 1$ there exist uniquely determined integers $a \gt 0$ and $b \gt 0$ such that $n = a^2b$ where $b$ is square-free.
negative_passages.docid: 98440
negative_passages.text: Show that any non-negative rational integer may be written as the sum of two squares if it is of the form....