query: numbers x such that the sum of the divisors is a perfect square
positive_passages.docid: 152754
positive_passages.text: When is the sum of divisors a perfect square?
negative_passages.docid: 78593
negative_passages.text: Prove that the product of four consecutive positive integers plus one is a perfect square
negative_passages.docid: 152643
negative_passages.text: $(xy + 1)(yz + 1)(zx + 1) $ is a perfect square if and only if each factor is
negative_passages.docid: 142513
negative_passages.text: Prove that $2xy\mid x^2+y^2-x$ implies $x$ is a perfect square.
negative_passages.docid: 134
negative_passages.text: Prove that the expression is a perfect square
negative_passages.docid: 45911
negative_passages.text: Prove that none of $\{11, 111, 1111,\dots \}$ is the perfect square of an integer
negative_passages.docid: 79103
negative_passages.text: Check if a number is a perfect square
negative_passages.docid: 100913
negative_passages.text: It is impossible for $(x-1)^2+x^2+(x+1)^2$ to be 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: 22800
negative_passages.text: Can $\sqrt{n} + \sqrt{m}$ be rational if neither $n,m$ are perfect squares?
negative_passages.docid: 155784
negative_passages.text: If $\gcd(a, b) = 1$ and if $ab = x^2$, prove that $a, b$ must also be perfect squares; where $a,b,x$ are in the set of natural numbers
negative_passages.docid: 41789
negative_passages.text: Fermat's theorem on sums of two squares
negative_passages.docid: 101832
negative_passages.text: Proof that sum of first $n$ cubes is always a perfect square
negative_passages.docid: 35326
negative_passages.text: There are infinitely many $N$ such that $\frac{N}{2}$ is a perfect square, $\frac{N}{3}$ is a perfect cube, and $\frac{N}{5}$ is a perfect fifth power
negative_passages.docid: 149071
negative_passages.text: Let $d$ be any positive integer not equal to $2, 5,$ or $ 13$ , then $\exists a, b \in \{2, 5, 13, d\}$ such that $ab − 1$ is not a perfect square?
negative_passages.docid: 10920
negative_passages.text: $\delta_2(n)$ the sum of the squares of the positive divisors of $n$.
negative_passages.docid: 59782
negative_passages.text: Are there any $n$ for which $ n^4+n^3+n^2+n+1$ is a perfect square?