query: How to prove that $x^2+1=5^y$ has no positive integer solutions for $y\geq 2$?
positive_passages.docid: 20705
positive_passages.text: On equations $m^2+1=5^n$
negative_passages.docid: 25078
negative_passages.text: What is the solution set for the equation$ x ^ y = y ^ x$ in positive real numbers, x not equal to y?
negative_passages.docid: 1001
negative_passages.text: Number of integer solutions of $x^2 + y^2 = k$
negative_passages.docid: 3036
negative_passages.text: Is a polynomial equation of degree $\ge 5$ not solvable by any way?
negative_passages.docid: 59653
negative_passages.text: $x^2-y^2=2s$, s cannot be an odd integer
negative_passages.docid: 121658
negative_passages.text: $x^2+y^2-1$ irreducible in $\mathbb{Q}[x,y]$?
negative_passages.docid: 28274
negative_passages.text: Prove that $x^2 + y^2 = 3(z^2 + m^2)$ has no solutions in integer
negative_passages.docid: 99000
negative_passages.text: No polynomial with integer coefficients satisfies the conditions
negative_passages.docid: 146126
negative_passages.text: Infinite positive integer solutions of the equation: $x^2+x+1=(y^2+y+1)(n^2+n+1)$
negative_passages.docid: 109532
negative_passages.text: Proving $5^n \equiv 1 \pmod {2^r}$ when $n=2^{r-2}$
negative_passages.docid: 74920
negative_passages.text: Prove that the equation $x^2-y^2 = 2002$ has no integer solution
negative_passages.docid: 46668
negative_passages.text: How to find integer solutions to $M^2=5N^2+2N+1$?
negative_passages.docid: 68233
negative_passages.text: Fermat's little theorem's proof for a negative integer
negative_passages.docid: 29780
negative_passages.text: If $x$ is a positive rational but not an integer, is $x^x$ irrational?
negative_passages.docid: 144765
negative_passages.text: Solving $\sqrt{x+5} = x - 1$
negative_passages.docid: 148243
negative_passages.text: Prove that there are no positive integers $x$ and $y$ such that $x^3 + y^3 = 10^3$.
negative_passages.docid: 155363
negative_passages.text: Find all integers $(x, y)$ such that $1 + 2^x + 2^{2x + 1} = y^2$