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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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An equation has infinitely many positive integers solutions

We know that the Diophantine equation: $ax + by = c$, has infinitely many integer solutions if $\gcd(a,b)|c$. Now, I am asking about the case where this equation has infinitely many positive integers solutions. My solution: Assuming $a>0$ and…
Safwane
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Odd cubes of the form $x^2+y^2-xy$

Statement: If $x,y,n$ are odd integers such that $\gcd(x,y)=1$ and $x^2+y^2-xy=n^3$, then $x+y=2c^3-2cd^2$, where $c,d$ are co-prime integers. Two examples for clarity: $x=-71, y=181$ $(-71)^2+181^2-(-71)(181)=37^3$ And…
Joseph
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Odd factors of $a^2+3b^2$ have this same form, when $\gcd(a,b)=1$

edit: I originally made this post to check my understanding of this proof. Now I understand this is not a good type of question to ask. Despite regretting making this post originally, I would rather fix this proof than delete it if that's ok. My…
Joseph
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General Solution of Diophantine equation

Having the equation: $$35x+91y = 21$$ I need to find its general solution. I know gcf $(35,91) = 7$, so I can solve $35x+917 = 7$ to find $x = -5, y = 2$. Hence a solution to $35x+91y = 21$ is $x = -15, y = 2$. From here, however, how do I move on…
MrD
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Finding $1998x + 5y + 3z$ given $\frac{1}{x} + \frac{1}{y} = \frac{3}{z}$

Suppose $x,y,z$ are positive integers so that $y$ and $z$ are prime and $x = yz.$ If they satisfy $$\frac{1}{x} + \frac{1}{y} = \frac{3}{z},$$ find $1998x + 5y + 3z.$ I first moved $\frac{1}{y}$ over to the other side and expanded out, but from…
questionasker
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Prove that $x^2 + 8 = 3^y$ has only one solution: $x=1, y=2,$ where $x,y\in \mathbb{N}$

So I've been practicing some Diophantine equations, but this is the first one where the power is a variable, I don't even know how to begin. Prove that $x^2 + 8 = 3^y$ has only one solution: $x=1, \ y=2,$ where $x,y\in \mathbb{N}$
Billaam Ben Beor
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Euler conjecture for 514 case

I was trying to find counterexamples for the Euler conjecture for fifth power, but I couldn't find any aside from $27^5+84^5+110^5+133^5=144^5.$ Python script finds the first solution using a brute force approach in a second. It takes one day to…
Stepan
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Simple looking diophantine equation...

I recently found myself asking if the following (diophantine) expression ever evaluates to a square number: $$5+12n$$ I was surprised both to be unable to stumble across an integer value for $n$ that results in a square number, and then surprised…
aSteve
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An asymptotic from below to number of solutions to $xyz + x + y = n$

Let $n$ be a positive integer. One can show (not that easy, but still via elementary methods) that the number of triples $(x,y,z)$ of positive integers satisfying $xyz + x + y = n$ is $O(n^{\frac{1}{3}+\varepsilon})$ for any $\varepsilon > 0$. (Use…
DesmondMiles
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solution of equation $x^n-y^m=2$

can anyone give me solution of $x^n-y^m=2$ where $x,y$ are positive integers and $m,n$ greater than or equal to $2$ except $(x,y,n,m)=(3,5,3,2)$ ?if this is only solution of given equation,then can anyone prove it ?
Krishna Mishra
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What are the integer solutions of the equation $(x^2 -1)(y^2 -1)=2(7xy-24)$

Can you help with this one? I've been trying fruitlessly for hours $$(x^2 -1)(y^2 -1)=2(7xy-24)$$
justanotherguy
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2018 Diophantine Equation

Problem: Find all ordered pairs of positive integers $(a,b)$ such that $\frac 1a + \frac1b = \frac 3{2018}$. My beginning solution: Using basic algebra and factorization, we can see that the equation reduces to $$2 \cdot 1009 \cdot (a+b) = 3 \cdot…
L. Tim
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Solution to a Third degree diophantine equation

I have two diophantine equations of the third degree viz.$$2b_1^3l_1+3b_1^2l_1^2+b_1l_1^3=k$$ and $$2b_2^3l_2+3b_2^2l_2^2+b_2l_2^3=k$$ The aim is to find distinct values of $(l_i,b_i)$ which satisfy this solution. For example both $(3,2)$ and…
RTn
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Solve the equation below for all m and n which are positive integers

Find all positive integers $m$ and $n$ such that $$m^2+2\cdot 3^n=m(2^{n+1}-1).$$
FC Barcelona
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Error in finding Solution for positive Diophantine equation

I need to find minimum value of c above which there always exists a non-negative solution for the equation $$4x + 7y = c$$ I tried using Diophantine equation but I am not able to find the mistake in my approach, could someone please point out?…
Rishabh
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