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Questions tagged [diophantine-equations]

Use for questions about finding integer or rational solutions to polynomial equations.

Use this tag for questions about finding integer, or perhaps rational, solutions to polynomial equations.

Diophantine equations are named after Diophantus of Alexandria, a third century Greek mathematician.

An example of a Diophantine equation is to find all quadruples of integers $(w,x,y,z)$ such that $$w^2+x^2=3(y^2+z^2).$$

Solving Diophantine equations often involves other areas of mathematics such as congruences, linear algebra, inequalities, forms (e.g., binary quadratic forms), and elliptic curves. Special solution methods include comparing divisors, considering orders of magnitude, Fermat's method of descent, and finding intersections of curves with lines of rational slope through a known rational point.

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Corollary to the Rational Distance Problem; Finding a certain point in the plane satisfying certain properties

Does anyone know a point in the 2D plane which satisfies the following properties: Its distance from two vertices of a unit square is rational. Its distance from other two vertices of a unit square is irrational. It does not lie on any lines of…
Agile_Eagle
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prove that $\sum_{i+j+k=p}{p\choose i+j+k}x^iy^jz^k\equiv x^p+y^p+z^p\pmod{(x+y)(y+z)(z+x)}$

Let $x,y,z$ be integers and $p$ an odd integer.How can I prove that $$\sum_{i+j+k=p}{p\choose i+j+k}x^iy^jz^k\equiv x^p+y^p+z^p\pmod{(x+y)(y+z)(z+x)}$$
user97615
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Diophantine equation: $\frac{1}{a^2}+\frac{1}{b^2} =\frac{1}{c^2}$

Diophantine equation: $\frac{1}{a^2}+\frac{1}{b^2} =\frac{1}{c^2}$ What is sum of all a equal or less than $100$?
Weiqing Wu
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How to solve $xy=5x+5y$ for integer solutions

I did a bit of research and as I understand it would probably involve Diophantine equations. Unfortunately I have no idea where to start. Any help would be greatly appreciated. Thanks ^^
MadRabbit
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What are all the two-digit positive integers in which the difference between the integer and the product of its two digits is $12$?

What are all the two-digit positive integers in which the difference between the integer and the product of its two digits is $12$ ? What I did so far: $10a+b-ab=12$, $10a+b(1-a)=12$, $-10(1-a)+b(1-a)=2$, $(b-10)(1-a)=2$ Then I solve and get…
jonyoung2002
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Prove for the following set of Diophantine equations.

Prove that there are no integers $q$, $m$, $n$ and $p$ for the following Diophantine equations:- $$ 7m^2 = 1 + q^2$$ $$p^2 - 1 = 7n^2$$
Lucifer -
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Solving diophantine equation with coefficients

We know this equation : $$ x_1 + x_2 + \dots + x_n = c$$ and also the number of solutions is obvious . But how we can understand number of solutions in this equation : $$ a_1x_1 + a_2x_2 + \dots + a_nx_n = k$$ Please Help!
S.H.W
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Diophantine equation no integer solutions

Show that the following equation has no integer solutions: $x^3+3x^2+2x=z^3-4z+4.$ No idea where to start because it has no $y$ functions. Also I need to find the integer solutions to $y^2+x^2=9-z^2$.
K_uddin
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What are all the solutions to ax + by = 0 with nonzero integer coefficients?

Suppose we have an equation of the form $ax+by=0$ with $a,b,c \in \mathbb{Z}$. For simplicity, $a \neq 0, b \neq 0$. Then, a single solution to this equation is $(x_0, y_0)=(-a, b)$. My book states that all solutions are of the form $(x_0,…
www.data-blogger.com
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Diophantine Algorithm Word Problem 1

A says, "We three have P100 altogether". B says, "Yes, and if you had six times as mich and I had one third as much, we three would have still have P100". C says, "It's not fair. I have less than P30". Who has what?
Issa
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Finding integer solutions to $[\sqrt{x^2+y^2}]=k$

I am in the process of trying to devise analytic geometric methods for digital geometry, and came across the following equation: $[\sqrt{x^2+y^2}]=5$, where $[x]$ denotes conventional rounding. Apart from (obviously) noting that $0\le x,y\le5$, I…
Ethan Hunt
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Equations $n^{am+bn}=m^{cn+dm}$

I will be very grateful, if someone show me, how to solve such equations. Example 1. $$ n^{m+2n}=m^{4n} $$ n,m - positive integers. Thanks a lot.
user16023
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Quadratic Diophantine Equation with Rational Coefficients

The problem is as below: Solve all solutions to $x^2+\dfrac{p}{q}(xy)+y^2=z^2$ for $x$, $y$, $z\in\mathbb{Q}$ and $p$, $q\in\mathbb{N}$ with $\gcd{(p,q)}=1$. My attempt: Noticing that for a Diophantine Equation $x^2+axy+y^2$, it's solution is…
blastzit
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Find all triplets $(x,y,p)$ such that $\frac{xy^3}{x+y}=p$

Find all triplets $(x,y,p)$ such that $\frac{xy^3}{x+y}=p$, where $x$ and $y$ are positive integers and $p$ is prime. I don't have much experience with Diophantine equations such as this, and I don't know how to fully solve it. After a bit of…
zagaluke
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Find all possible three digit numbers whose digits' sum equals 12

How many 3 digit numbers exist such that the sum of their digits equals 12? I ran a little program and found that there are $66$ of such numbers. I feel that this type of problem is similar in style to that of the Diophantine equation $ax+by=c$…
Ian L
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