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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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Text problem (linear diophantine equation??)

An intermediary bought $\,x\,$ units of a commodity at a price of $\,€27\,\,$ and sold $\,y\,$ units at a price of $\,€37\,\,$ on $\,4$ April this year. In doing so, he made a profit of $\,€89\,\,$ earned. The following task is given for this…
user1159827
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Who can prove that $7|4x^2+5x+2+d $ following solutions I found are correct?

Who can prove that $7|4x^2+5x+2+d $ following solutions I found are correct (complete)? Where $d$ is integer constant (not relation with $x$), find $d$ which make $7|4x^2+5x+2+d $ have solution, and give corresponding solution $x$ as well. I have…
xMath
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Solving a diophantine equation involing squares and a prime number

I have the following diophantine equation I wish to find and prove that there is ONLY ONE solution to the following equation $x^2=y^2 + 31$ I understand that 31 is a prime number thus we can equate $x^2 - y^2$ is a perfect square therefore we can…
Khandkar Islam
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Given integers $a$ and $b$, what integers $k$ make $ak+b$ a power of 2?

Recreationally, I have been studying Fermat numbers and have been trying to come up with a construction to produce arbitrarily large composite Fermat numbers. It's been leading me to many Diophantine equations, which are a new and interesting topic…
Graviton
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Strategy To Solve Diophantine Equations

Find all integers n for which the equation $$x^3+y^3+z^3-3xyz=n$$ is solvable in positive integers.
Yooooooooooo
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Find all positive integer solutions verifying two conditions

Let us consider the following Diophantine problem: Find all positive integer solutions verifying the two conditions: (1) $(x+1)^2$ is a multiple of $2^{y}$ (2) $2^{y}≤x<2^{y+1}$ where $y$ is a fixed positive integer. Context of the question: Let us…
Safwane
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Transform quadratic ternary form to normal form

Does anyone know of an integral transform which transforms the normal form $Ax^2 + By^2 + Cz^2 + Dxy + Eyz + Fzx = 0$ to the form $ax^2 + by^2 + cz^2 = 0$ ? Thanks in advance.
thilinarmtb
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How the Simplex method can be applied in solving the following system of Linear equations

How the Simplex method can be applied in solving the following system of Linear equations? \begin{align*} x_1–x_3+4x_4 &=3 \\ 2x_1–x_2&=3 \\ 3x_1–2x_2–x_4&=1 \end{align*} Where $$ x_1,\;x_2,\;x_3,\;x_4\geq0 $$ What would be the proposed Objective…
user86306
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Diophantine Equation solved with elliptic curves

I want to know how to find all solutions in $\mathbb{Z}$ for $$ 2a^2 -3ab +5c^2 =0. $$ I already solved it and I will post my solution soon. One solution for example is $(15,11,3).$
calculatormathematical
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Method to solving $a^n + b^n = c$

I have been wondering for a while if there is a method to solving the following form of equation $a^n + b^n = c$ Where a,b,c are all integers. For example, $2^n + 5^n = 29$. One can quickly see that n is 2 but is there a general method?
user997889
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The diophantine equation $\frac{1}{a}+\frac{1}{b}=\frac{n}{a+b}$

How do I solve This diophantine equation $\dfrac{1}{a}+\dfrac{1}{b}=\dfrac{n}{a+b}$ of unknown $(a,b,n)\in \mathbb{(N^*)^3}$.
user1047205
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About a quartic diophantine equation in several variables

I have a nonlinear diophantine equation of the form $$g(x,y)t^4+h(x,y)t^3+w(x,y)t^2+f(x,y)t+d(x,y)=0$$ such that $t$ is a positive integer variable and $g(x,y),h(x,y),w(x,y),f(x,y),d(x,y)$ are polynomial functions in the positive integers variables…
Safwane
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Positive integral solutions $(m,n)$ to $m = n^m$

Are there any positive integral solutions $(m,n)$ to the diophatine equation $n=m^{n}$ besides $(m,n)=(1,1)$? Not sure how to approach this question. I got the (obvious) solution by guessing. It seems clear that $m\leq{n}$.
student
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Diophantine equation: $x^2-y^2=z$ for a fixed integer $ z$

Is there any relatively efficient way to calculate integer solutions $x$ and $y$ to the equation $x^2 - y^2 = z$ for a fixed integer $z$? May or may not be useful: $z$ is an odd composite number Thanks
user2505634
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Solving a linear Diophantine equation where c is a multiple of $7$ (or any other prime)

I want to solve $ax + by = 7k$ (a multiple of 7) with all letters being integers. For example: $6x + y = 7k$. With Wolfram Alpha I found one parameterization of the solution space: $y = 7n - 6x$ ($n \in \mathbb{Z}$). But somehow I found another…
Rüdi Jehn
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