query: Prove that there are no natural numbers to satisfy the equation.
positive_passages.docid: 28274
positive_passages.text: Prove that $x^2 + y^2 = 3(z^2 + m^2)$ has no solutions in integer
negative_passages.docid: 43939
negative_passages.text: Show that $3^{2n+1}-4^{n+1}+6^n$ is never prime for natural n except 1.
negative_passages.docid: 145519
negative_passages.text: Prove that for any natural number $n>1$, we have $(2n)!<(n(n+1))^n$
negative_passages.docid: 33757
negative_passages.text: For what natural numbers is $n^3 < 2^n$? Prove by induction
negative_passages.docid: 106271
negative_passages.text: Solutions of Diophantine equations in Natural numbers
negative_passages.docid: 121469
negative_passages.text: Proof of observations on natural numbers being expressed as differences of squares.
negative_passages.docid: 101169
negative_passages.text: Prove that $x^2 = - 1$ has no solution in $\mathbb{Z}$
negative_passages.docid: 46946
negative_passages.text: Is there an elementary proof that $\sum \limits_{k=1}^n \frac1k$ is never an integer?
negative_passages.docid: 36011
negative_passages.text: Show that the product of two consecutive natural numbers is never a square.
negative_passages.docid: 16971
negative_passages.text: Prove that no ordinal is an element of itself
negative_passages.docid: 33705
negative_passages.text: Is $0$ a natural number?
negative_passages.docid: 73262
negative_passages.text: Prove that the equation $\tan (z)=z$ has only real roots.
negative_passages.docid: 4530
negative_passages.text: For which natural numbers are $\phi(n)=2$?
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: 106374
negative_passages.text: Show that $2x^6+12x^5+30x^4+60x^3+80x^2+30x+45=0$ has no real roots
negative_passages.docid: 144802
negative_passages.text: Prove no solution to $x^2\equiv3\bmod5$
negative_passages.docid: 108748
negative_passages.text: Why there are real numbers with infinite digits, but no such natural numbers (or another reason why real numbers are uncountable)