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### Find all integer solutions for the equation $|5x^2 - y^2| = 4$

In a paper that I wrote as an undergraduate student, I conjectured that the only integer solutions to the equation $$|5x^2 - y^2| = 4$$ occur when $x$ is a Fibonacci number and $y$ is a Lucas number. ...
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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 $... 6answers 2k views ### How to find solutions of$x^2-3y^2=-2$? According to MathWorld, Pentagonal Triangular Number: A number which is simultaneously a pentagonal number$P_n$and triangular number$T_m$. Such numbers exist when $$\frac{1}{2}n(3n-1)=\frac{1}{... 5answers 1k views ### Another quadratic Diophantine equation: How do I proceed? How would I find all the fundamental solutions of the Pell-like equation x^2-10y^2=9 I've swapped out the original problem from this question for a couple reasons. I already know the solution to ... 5answers 379 views ### If (m,n)\in\mathbb Z_+^2 satisfies 3m^2+m = 4n^2+n then (m-n) is a perfect square. I came across this question on another forum. The question is:$$ \text{If$m,n\in \mathbb{Z}_+$such that$3m^2+m=4n^2+n$, then$(m-n)$is a perfect square.}$$I have managed to partially prove ... 4answers 1k views ### Does the Pell-like equation X^2-dY^2=k have a simple recursion like X^2-dY^2=1? If d \ne 0 is a non-square integer, and (u,v) is an integer solution to the Pell equation$$ X^2 - dY^2 = 1, \tag{$\star$} $$then each solution (x_i,y_i) can be recursively calculated using ... 6answers 2k views ### Positive integer n such that 2n+1 , 3n+1 are both perfect squares How many positive integer n are there such that 2n+1 , 3n+1 are both perfect squares ? n=40 is a solution . Is this the only solution ? Is it possible to tell whether finitely many or ... 2answers 126 views ### Solve the following equation for x and y: x^2 = y^2 + xy + 5, where x and y are natural numbers. Here is what I have so far: x \neq y (from the equation). x is always odd (using the equation and assuming 2 cases - y is odd or ... 2answers 275 views ### Maps of primitive vectors and Conway's river, has anyone built this in SAGE? I am attempting to teach number theory from John Stillwell's Elements of Number Theory in the upcoming semester. There are two sections (5.7 and 5.8) which describe the diagrammatic method for the ... 2answers 1k views ### Finding all solutions of the Pell-type equation x^2-5y^2 = -4 I wanted to solve the equation x^2-5y^2 = -4 with x and y integers. Let \omega=\frac{1+\sqrt5}{2} and A = \mathbb{Z}[\omega]. One can reduce the Pell equation to finding the elements of A ... 1answer 317 views ### Why can't the Alpertron solve this Pell-like equation? Dario Alpern's Alpertron is convenient for solving Pell and Pell-like equations. It can even solve the one at the heart of Archimedes' cattle problem,$$p^2-(4)(609)(7766)(4657^2)q^2=1$$and give ... 3answers 254 views ### Find all natural numbers n such that 21n^2-20 is a perfect square. Find all natural numbers n such that 21n^2-20 is a perfect square. I have got the following solutions via programming: n=1,2,3,9,14,43,67 but how can I find these manually? How can I ... 2answers 133 views ### Integer as sum of 6 squares. Can every integer be written as sum of exactly 6 squares? I am also curious to know when an integer can be written as sum of exactly 8 squares. I know the problem is related to Waring-Hilbert ... 1answer 743 views ### how to solve binary form ax^2+bxy+cy^2=m, for integer and rational (x,y) solve 3x^2+3xy-5y^2=55 using number theory tools ,i have found the following \Delta=3^2+4(5)(3)=9+60=69 d=69,u=1 w_{69}=\frac{1+\sqrt{69}}{2} O_{69}=\theta_{-11}=[1,\frac{1+\sqrt{69}... 2answers 145 views ### Infinitely many systems of 23 consecutive integers Prove that there are infinitely many systems of 23 consecutive integers whose sum of squares is a perfect square. My try:$$(n-11)^2+\cdots+(n+11)^2=23n^2+1012=23(n^2+44)=m^2$$so$m=23k$,$n^2=...

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