query: Method of ascent to prove that $x^2 − 3y^2 = 1$ has infinitely many solutions
positive_passages.docid: 77763
positive_passages.text: Proof that the equation $x^2 - 3y^2 = 1$ has infinite solutions for $x$ and $y$ being integers
negative_passages.docid: 137308
negative_passages.text: If an IVP does not enjoy uniqueness, then it possesses infinitely many solutions
negative_passages.docid: 130565
negative_passages.text: Find all solutions to $x^3+(x+1)^3+ \dots + (x+15)^3=y^3$
negative_passages.docid: 1189
negative_passages.text: Using Fermats prime numbers to prove that there is infinitely many prime numbers
negative_passages.docid: 64731
negative_passages.text: Determine $"a"$ for which the system has no solutions, infinitely many solutions, or has one solution
negative_passages.docid: 62161
negative_passages.text: Solutions of the Pell-type equation $x^2-2y^2=-1$
negative_passages.docid: 133101
negative_passages.text: Number of solutions of $P(x)=e^{ax}$ if $P$ is a polynomial
negative_passages.docid: 94407
negative_passages.text: Number of solutions for $x[1] + x[2] + \ldots + x[n] =k$
negative_passages.docid: 133925
negative_passages.text: Find All Solutions to System of Congruence
negative_passages.docid: 127945
negative_passages.text: Number of solutions to the congruence $x^q \equiv 1 \mod p$.
negative_passages.docid: 16946
negative_passages.text: How many solutions for an equation with simple restrictions
negative_passages.docid: 158933
negative_passages.text: Rational solutions of $y^2 = x^3 - x$
negative_passages.docid: 71108
negative_passages.text: Is there an integer $N>0$ such that $\varphi(n) = N$ has infinitely many solutions?
negative_passages.docid: 47247
negative_passages.text: How do we prove the existence of uncountably many transcendental numbers?
negative_passages.docid: 55197
negative_passages.text: Roots of $ x^3-3x+1$
negative_passages.docid: 6484
negative_passages.text: Polynomials in one variable with infinitely many roots.
negative_passages.docid: 82571
negative_passages.text: Number of solutions of $x^2=1$ in $\mathbb{Z}/n\mathbb{Z}$