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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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Diophantine equation = c?

I'm used to solving the most basic equations not specifying the c, but now I have 7106x + 4320y = 6 And I don't know how to calculate the 6
Manumit
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No integer $x$ such that $(x-y)^3+ x^3 = (x+y)^3$

It seems there is no integer $x$ such that such that $(x-y)^3+ x^3 = (x+y)^3$ where $y$ is a non-zero integer. At least I can't find one. Am I right and if so, how can one show it?
user66307
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Examples of finding algebraic solutions to Diophantine equations.

I saw to prove that $a^2+b^2=c^2$ you can produce the algebraic solutions $a = x^2-y^2, b = 2xy, c = x^2+y^2$. To check that works just multiply it out but it produces infinitely many solutions when you set x and y to be integers. Can you please…
quanta
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How to prove every term of this sequence is not a natural number

Sorry for the repost and for my "bad" English. I made a lot of errors in the previous one, so here's my actual question: Let's take a look at this sequence: (1) $[a_1,a_2,a_3,a_4,...,a_x]$ where $$a_1=\frac{3 \cdot…
Andrew
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Finding all solutions: $a^2 + b^2 = c^2 + d^2$

I want to find all solutions to the problem of two squares equaling two other squares. $$a^2 + b^2 = c^2 + d^2 \qquad b \le N$$Clearly, without loss of generality, I can assume that $$gcd(a,b,c,d) = 1$$ and $c\le a \le b \le d$. But after that, I'm…
amcalde
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algorithm for positive integer solutions of equation $a^3+b^3=22c^3$

This is a look-a-like to Fermat's last theorem for $n=3$, but it has solutions! I believe that its solution requires knowledge of the techniques of algebraic or analytic number theory which I don't have.
mike brady
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Find all integer solutions of equality

Find all integer solutions of equation $$x^3+(x+1)^3+...+(x+7)^3=y^3$$ I've solved it by opening brackets and consideration of signs but I think there is simpler way of solving it .
Antony Tito
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How can I intuitively understand the algorithm for finding the integer solutions to $ax+by=c$?

Recently I've started to take interest in linear diophantine equations (they play a key role in a math puzzle I stumbled upon). I don't have a strong math background, and at first I had no clue how to solve an equation like $ax+by=c$ over the…
JLagana
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Does $~4^y+1=4xy^2+x~$ have infinitely many solutions in integers?

Consider the equation : $~4^y+1=4xy^2+x$ I have found that this equation has integer solutions for following values of $~y$ : $y\in \{1,2,193,10068,29570,..\}$ Question : Are there finitely or infinitely many solutions ? Can one in practice compute…
Pedja
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Find Solutions To Some Diophantine Equations

I would like to find the solutions to $$i)\qquad a(a+b+c)=bc \\ii)\qquad a(a+b+c)=2bc \\iii)\qquad a(a+b+c)=3bc$$ for $0< a \le b \le c$ and of course: $\textrm{gcd}(a,b,c) = 1$ (since those are the interesting cases). To be honest, Diophantine…
amcalde
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Diophantine equation with bounded variables

I was reading the follow paper about cryptography: Fully Homomorphic Encryption over the Integers, but I faced a proof of Lemma which I didn't undestand: the Lemma A.1. In short, there is a proposition that says that "$x$ can be write as $p\alpha +…
Hilder Vitor Lima Pereira
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Prove there are no non-trivial solution to $3x^2 - 5y^2 + 7z^2 = 0$

I've tried using modulo $3$, and I get it down to $y^2 + z^2 = 0 \pmod 3$ ; I don't know where to go from here though. I justified my answer by stating that, because we're in $\pmod 3$ and we need non-trivial solutions, the only solutions…
Lerbi
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How many natural solutions does the equation $x^2 - c y^2 = 1$ have?

How many natural solutions has equation $x^2-cy^2=1$ depends on value of $c$ . I think I've seen this problem somewhere as a theorem but I can't remember where .
Antony
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How to solve the system $\frac{35-12b}{a-b}= \frac bk,\frac{12a-35}{a-b}= \frac {a}{1-k}$

I am trying to solve the system of diophantine equations: $$ \begin{align*} \frac{35-12b}{a-b} &= \frac bk \\[6pt] \frac{12a-35}{a-b} &= \frac {a}{1-k} \end{align*} $$ Where $a-b\ne 0,$ and $ k\,(1-k) \ne 0$ . I know for fact that the solutions…
user97615
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When is the equation $x^2-d^n y^2 = -1$ solvable?

My goal is to prove or disprove that if $x^2-dy^2=-1$ is solvable, then $x^2-d^ny^2 = -1$ is solvable for every odd $n \geq 1$. I do know that the former is solvable if and only if the continued fraction of $\sqrt{d}$ has an odd period, and I feel…
maudestly
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