query: Prove that either $(2^{10500} + 15)$ or $(2^{10500} + 16)$ is not a perfect square.
positive_passages.docid: 147666
positive_passages.text: How to prove that either $2^{500} + 15$ or $2^{500} + 16$ isn't a perfect square?
negative_passages.docid: 23523
negative_passages.text: Two problems on number theory
negative_passages.docid: 113359
negative_passages.text: Proving that $x^2 + 4$ is not divisible by $3$
negative_passages.docid: 88268
negative_passages.text: show $\frac{1}{15}< \frac{1}{2}\times\frac{3}{4}\times\cdots\times\frac{99}{100}<\frac{1}{10}$ is true
negative_passages.docid: 54355
negative_passages.text: $1+x+\ldots+x^n$ perfect square
negative_passages.docid: 100399
negative_passages.text: A doubt with a part of a certain proof.
negative_passages.docid: 77240
negative_passages.text: Show that $4 + \sqrt{10}$ is not prime in $\mathbb{Z}\bigl [\sqrt{10}\bigr]$
negative_passages.docid: 143772
negative_passages.text: How can I algebraically prove that $2^n - 1$ is not always prime?
negative_passages.docid: 6582
negative_passages.text: Prove the existence of the square root of $2$.
negative_passages.docid: 67356
negative_passages.text: When is $2^x+3^y$ a perfect square?
negative_passages.docid: 157287
negative_passages.text: Is there a simple method to prove that the square of the Sorgenfrey line is not normal?
negative_passages.docid: 102154
negative_passages.text: If $a$ and $b$ are consecutive integers, prove that $a^2 + b^2 + a^2b^2$ is a perfect square.
negative_passages.docid: 59653
negative_passages.text: $x^2-y^2=2s$, s cannot be an odd integer
negative_passages.docid: 47471
negative_passages.text: The square of n+1-th prime is less than the product of the first n primes.
negative_passages.docid: 153594
negative_passages.text: When is $1^5 + 2^5 + \ldots + n^5$ a square?
negative_passages.docid: 10882
negative_passages.text: irrationality of $\sqrt{2}^{\sqrt{2}}$.
negative_passages.docid: 86285
negative_passages.text: Given positive integers $ x , y$ and that $3x^2 + x = 4y^2 +y$ Prove that $x-y$ is a perfect square