2
votes
2answers
185 views

number of solutions of x*y<N [closed]

How many solutions (unique pairs (x,y) ) exist for equation $xy < N$ ? constraints : $x >1 , y>1 , N<=50000$ I tried following method , but it fails for say N=24 , in which i calculate ...
2
votes
0answers
56 views

An algorithm for solving linear diophantine equations?

I am entering an interesting team based math contest called the purple comet, and quite a lot of questions on this contest involve Diophantine equations. For this contest, you are given a computer, ...
0
votes
0answers
59 views

Finding all integral solutions of a positive definite quadratic equation

Let $q(x_1,\ldots,x_n)$ be an integral positive definite quadratic form. For $d\in\mathbb{N}$ the equation $$q(x_1,\ldots,x_n)=d$$ has a finite number of integral solutions. Is there an algorithm to ...
2
votes
2answers
62 views

Linear Diophantine equation in two variables with additional constraints

Given, $$aX + bY = c$$ where, $$c > b > a > 0;\quad X, Y > 0;\quad b\nmid c, a\nmid c$$ I want to find out if a solution exists as efficiently as possible (I'm not interested ...
1
vote
1answer
93 views

How to count solutions to a linear Diophantine equation?

I have the following linear Diophantine equation. $a+5b+20c+50d+100e+200f=300$ I wish to know the number of solutions to this equation with the constraint that $a,b,c,d,e,f \ge 0$. What algorithms ...
3
votes
3answers
446 views

Heronian triangle Generator

I'm trouble shooting my code I wrote to generate all Heronian Triangles (triangle with integer sides and integer area). I'm using the following algorithm $$a=n(m^{2}+k^{2})$$ $$b=m(n^{2}+k^{2})$$ ...
1
vote
2answers
363 views

Solution of Diophantine equation for polynomials

I'm trying to solve this polynomial Diophantine equation for $R$ and $S$: $ AR + BS = G $ where $A(x) = a_0x^{n_a} + a_1x^{n_a-1} + a_2x^{n_a-2} + ... + a_{n_a}$ $B(x) = b_0x^{n_b} + b_1x^{n_b-1} + ...
1
vote
1answer
107 views

Diophantine equations of the form $xy = n$

I am looking for a general way to estimate the number of possible solutions to solve the diophantine equation of the form $xy = n$, where $n$ is a positive integer. Note that $n$ can be an unusually ...
3
votes
2answers
426 views

Algorithm to determine if a Diophantine Equation has an infinite number of solutions

In their paper , Marker and Slaman, proved the decidability of the the theory of the natural numbers with the quantifier "for all but finitely many", One can obviously encode the question of whether ...
6
votes
1answer
171 views

How to study results of Diophantine equation?

I finally managed to learn a bit of number theory and Diophantine equation(with the help of Arturo Magidin's great answer in what type of math is this?). But I'm wondering what's the next step after? ...