query: How many sets of positive integers $(a,b,c,d)$ for $a^3+b^3 = c^3+d^3$?
positive_passages.docid: 153769
positive_passages.text: Numbers that can be expressed as the sum of two cubes in exactly two different ways
negative_passages.docid: 18693
negative_passages.text: If $a,b$ are positive integers such that $a^{n}+n$ divides $b^{n}+n$ for every $n$, then $a=b$
negative_passages.docid: 79568
negative_passages.text: Cardinality of the set $D$
negative_passages.docid: 106030
negative_passages.text: Find all the positive integer
negative_passages.docid: 128387
negative_passages.text: Given that $a+b\sqrt[3]{2} +c\sqrt[3]{4} =0$, where $a,b,c$ are integers. Show $a=b=c=0$
negative_passages.docid: 149489
negative_passages.text: 30 positive integers
negative_passages.docid: 56624
negative_passages.text: Prove that $0!+1! + 2! + 3! + ..... + n!$ $\neq$ $p^\text{r}$, where $n \geqslant 3$ and $n$, $p$ and $r$ are three integers
negative_passages.docid: 141093
negative_passages.text: How many real roots does $(x-a)^3+(x-b)^3+(x-c)^3$ have?
negative_passages.docid: 69874
negative_passages.text: Are there any integer solutions to $a^3=b^2$?
negative_passages.docid: 19046
negative_passages.text: Show that if $a$ and $b$ are positive integers with $(a,b)=1$ then $(a^n, b^n) = 1$ for all positive integers n
negative_passages.docid: 137010
negative_passages.text: What are integers $x$ and $n$ such as $ x^4+x^3+x^2+x+1 = n^2$?
negative_passages.docid: 45090
negative_passages.text: Given any positive real numbers $a,b,c$, we have $(a^{2}+2)(b^{2}+2)(c^{2}+2)\geq 9(ab+bc+ca)$
negative_passages.docid: 5287
negative_passages.text: Representing every positive rational number in the form of $(a^n+b^n)/(c^n+d^n)$
negative_passages.docid: 95490
negative_passages.text: What are the positive integer solutions to $x^2-x+1 = y^3$?
negative_passages.docid: 4750
negative_passages.text: Show that for any positive integer n, $(3n)!/(3!)^n$ is an integer.
negative_passages.docid: 23427
negative_passages.text: How to prove there are no more positive integers that are products of 2 and 3 consecutive numbers?
negative_passages.docid: 47737
negative_passages.text: Find all the possible values of $(a,b,c,d)$.