# Linked Questions

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### Minimum of $n$? $123456789x^2 - 987654321y^2 =n$ ($x$,$y$ and $n$ are positive integers)

What is the minimum of $n$? $x$,$y$ and $n$ are positive integers, find the minimum of $n$, such that: $123456789x^2 - 987654321y^2 =n$
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Consider the identity $(b-a)(b+a) = ab - 1$, where $a, b$ are nonnegative integers. We can also express this identity as $a^2 + ab - b^2 = 1$. This identity is clearly true when $a = F_{2i-1}$ and $... 3answers 1k views ### The quadratic diophantine$ k^2 - 1 = 5(m^2 - 1)$Here's the problem. Find the solutions of the following equation: $$k^2 - 1 = 5(m^2 - 1).$$ Here's my idea: The original equation can be written as: $$k^2 = 5m^2 - 4 \Longleftrightarrow k^2 - ... 2answers 835 views ### Solving the equation x^2-7y^2=-3 over integers I'd like to solve the following Pell equation:$$ x^2-7y^2=-3 $$Where x and y are integers. I applied the usual procedure, which avoids continued fractions: The two minimal positive integer ... 2answers 326 views ### Classification of the positive integers not being the sum of four non-zero squares It is well known that every positive integer is the sum of at most four perfect squares (including 1). But which positive integers are not the sum of four non-zero perfect squares (1 is still ... 2answers 319 views ### Maps of primitive vectors and Conway's river, has anyone built this in SAGE? I am attempting to teach number theory from John Stillwell's Elements of Number Theory in the upcoming semester. There are two sections (5.7 and 5.8) which describe the diagrammatic method for the ... 3answers 495 views ### Algebraic proof of non-trivial solution to the Pell's equation Let d be a square-free positive integer, and consider the pairs (x, y) \in \mathbb{Z}^2 that satisfy:$$x^2 - dy^2 = 1$$The existence of a non-trivial solution to this equation (i.e. distinct ... 1answer 1k views ### how to solve binary form ax^2+bxy+cy^2=m, for integer and rational (x,y) solve 3x^2+3xy-5y^2=55 using number theory tools ,i have found the following \Delta=3^2+4(5)(3)=9+60=69 d=69,u=1 w_{69}=\frac{1+\sqrt{69}}{2} O_{69}=\theta_{-11}=[1,\frac{1+\sqrt{69}... 3answers 204 views ### What are some books that are in the spirit of David A. Cox' “Primes of the Form x^2+ny^2” David A. Cox "Primes of the Form x^2+ny^2: Fermat, Class Field Theory, and Complex Multiplication." has a very good (at least to me, and many) methodology. He starts from page 1 asking a simple ... 1answer 412 views ### General method for determining if Ax^2 + Bx + C is square Is there a general method for solving Diophantine equations in the form Ax^2 + Bx + C = k^2, preferably turning them into Pell's equations, when possible? For example, 2x^2 + x + 1 = k^2 or 5x^2 +... 3answers 397 views ### Linear recurrence solution to Diophantine equation I have a Diophantine equation of the form:$$ax^2 + bx + c = y^2, \quad x, y \in \mathbb{Z^+}$$Is it true that there will always be a linear recurrence formula that generates all the solutions for x... 2answers 388 views ### Solutions to Diophantine Equations I am looking for integer solutions to the equation$$x^2 = 5y^2 + 14y + 1$$I know that Pell's Equation is of the form x^2 - ny^2=1 and that there exist algorithms to solve this equation. I was ... 2answers 173 views ### How to prove that the roots of this equation are integers? Let there be an equation a^2 + 4ab + b^2 - 121 = 0 where I want to prove that a,b are integers. Then I want to find whether there are integer values of b for which a is also an integer. Let us ... 1answer 331 views ### Modified Pell equation: x^2-D y^2 = m, m\neq1. How does one solve the Diophantine equation$$ x^2-Dy^2=m, $$where m is some fixed arbitrary integer? I understand that given the fundamental solution to r^2-D s^2=1, and any solution to the ... 1answer 187 views ### Can I check whether integral solutions exist if I know a rational solution? The pell-like equation$$x^2-101y^2=-71$$has the rational solution$(x,y)=(\frac{25}{2},\frac{3}{2})\$ Can I use this rational point to find out , whether an integral solution exists ? If yes, can ...

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