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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 $$\frac{1}{2}n(3n-1)=\frac{1}{... 3answers 2k views ### Find all integer solutions for the equation |5x^2 - y^2| = 4 In a paper that I wrote as an undergraduate student, I conjectured that the only integer solutions to the equation$$|5x^2 - y^2| = 4$$occur when x is a Fibonacci number and y is a Lucas number. ... 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 ... 3answers 288 views ### Is there a simple proof that if (b-a)(b+a) = ab - 1, then a, b must be Fibonacci numbers? [duplicate] 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 ... 5answers 984 views ### Generate solutions of Quadratic Diophantine Equation Recently I've asked a question for how to solve Quadratic Diophantine Equation and I got one interesting answer. Link to question: The quadratic diophantine  k^2 - 1 = 5(m^2 - 1) Here's the answer: ... 2answers 2k 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 ... 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 496 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$$The existence of a non-trivial solution to this equation (i.e. distinct ... 1answer 1k views ### how to solve binary form ax^2+bxy+cy^2=m, for integer and rational  (x,y) solve  3x^2+3xy-5y^2=55 using number theory tools ,i have found the following \Delta=3^2+4(5)(3)=9+60=69 d=69,u=1 w_{69}=\frac{1+\sqrt{69}}{2} O_{69}=\theta_{-11}=[1,\frac{1+\sqrt{69}... 3answers 204 views ### What are some books that are in the spirit of David A. Cox' “Primes of the Form x^2+ny^2” David A. Cox "Primes of the Form x^2+ny^2: Fermat, Class Field Theory, and Complex Multiplication." has a very good (at least to me, and many) methodology. He starts from page 1 asking a simple ... 1answer 415 views ### General method for determining if Ax^2 + Bx + C is square Is there a general method for solving Diophantine equations in the form Ax^2 + Bx + C = k^2, preferably turning them into Pell's equations, when possible? For example, 2x^2 + x + 1 = k^2 or 5x^2 +... 2answers 388 views ### Solutions to Diophantine Equations I am looking for integer solutions to the equation$$x^2 = 5y^2 + 14y + 1$$I know that Pell's Equation is of the form x^2 - ny^2=1 and that there exist algorithms to solve this equation. I was ... 2answers 173 views ### How to prove that the roots of this equation are integers? Let there be an equation a^2 + 4ab + b^2 - 121 = 0 where I want to prove that a,b are integers. Then I want to find whether there are integer values of b for which a is also an integer. Let us ... 2answers 151 views ### Infinitely many systems of 23 consecutive integers Prove that there are infinitely many systems of 23 consecutive integers whose sum of squares is a perfect square. My try:$$(n-11)^2+\cdots+(n+11)^2=23n^2+1012=23(n^2+44)=m^2$$so m=23k , n^2=... 1answer 187 views ### Can I check whether integral solutions exist if I know a rational solution? The pell-like equation$$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 ...
In general, what are the strategies for showing the completeness of a solution set for Diophantine Equations? For example, take the $\textit{Pell-type}$ equation $x^2 - dy^2 = a$. Say, you have a set ...
### Small integral representation as $x^2-2y^2$ in Pell's equation
Let $k$ be a "representable" positive integer, in the sense that $k=|x^2-2y^2|$ for some integers $x,y$. Does it necessarily follow that $k$ can also be represented with small parameters, i.e. \$k=|u^2-...