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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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Does the equation $a^2+d^2+4=b^2+c^2$ have any solutions?

Does the equation have $a^2+d^2+4=b^2+c^2$ where $d
Wesley
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Solve $an^2+bn+c=k^2$ over the integers.

I need to solve $5n^2+14n+1=k^2$ over integers $n$ and $k$. I was wondering if there is any general theory for solving diophantine equations of the form $an^2+bn+c=k^2$. For my specific case, I already found the first $21$…
SmileyCraft
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Solve the following Diophantine equation.

Find all integral pairs (x,y) such that - $$( xy - 1)^2 = (x +1)^2 + ( y+1)^2$$ My Approach : I just expanded this equation and wrote it in another form - $$\frac{(xy+1)(xy-1)}{(x+y)}-2=x+y$$ and from this we can say that $(x+y)|(xy+1) \…
Identicon
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Find solutions for $\lambda$ in this (diophantine?) equation?

I am interested in finding the solutions to the equation below: $$\lambda = \frac{2^c+3b}{2^d-3^a}$$ $$\lambda, a, b, c, d \in \Bbb Z^+$$ $$d>a,c$$ Edit: My original version was actually: $$\frac{3^a(\lambda) + 3(3b+2^c)+2^d}{2^e}=…
Roskiller
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On the Diophantine equation $ (12m^2 + 1)^x + (13m^2 - 1)^y = (5m)^z$

I have read a paper http://www.m-hikari.com/ija/ija-2015/ija-5-8-2015/p/teraiIJA5-8-2015.pdf concerning the Diophantine equation $$ (12m^2 + 1)^x + (13m^2 - 1)^y = (5m)^z$$ and the author have obtained one solution $(x,y,z)=(1,1,2)$ under the…
Theory Nombre
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Integer solutions to $x^y - y^x = 1$

The equation $x^y - y^x = 1$ has integer solutions $(2,1)$, $(3,2)$ and $(k, 0)$ (for any $k > 0$). Are there any others? Based on the graph (https://www.desmos.com/calculator/qyxoemixli, see below) it doesn't look like it, but is there a simple…
mweiss
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Find a solution to $x^2+y^2=65^2$ with $(x,y)=1$

Before this problem, I had to show that $65$ could be written as a sum of two squares. After this, I used ProofWiki to find solutions of the problem, which I got, for example, $65^2=(1+8^2)(4^2+7^2)=60^2+(-25)^2$. But $(60^2,25^2)\neq 1$. I used…
Hopeless
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General solution for nonlinear Diophantine equation

As the title says I'm looking for a general solution to a Diophantine equation of the form: $y^2 = x^2 + kx - m$ Where $x$ and $y$ are both positive integers. I know that $k$ and $m$ will always be a multiple of 2 if that helps. Here are a few…
CookieNinja
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Can $7n + 13$ ever equal a square?

If not, why not? Can it be proved? And if it can be proved that it does equal a square (which I doubt), what is the smallest value for which this occurs?
Dolphin
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Find all integer solutions to the equation $x^2 − x = y^5 − y$.

My professor gave us a worksheet with diophantine equations including this one he claims that it is one of the easier ones but that it has a unique solution. Find all integer solutions to the equation $$x^2 − x = y^5 − y$$. Also my professor will…
Jamie Lacey
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Integer solutions of $a^6 + 4 b^3 = 1$

This question is simplified case of Integer solutions of $a^6 + 4 b^3 = c^6$. Is there integer solutions for $$a^6 + 4 b^3 = 1$$ The way I am trying to prove is based on Fermat's Theorem, but I can not get the final result.
Gevorg Hmayakyan
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Prime numbers in equation

I am not able to solve this equation. Can anybody help? $p$, $q$ are prime numbers and $a$ is a positive integer. $$ \frac{pq}{p+q}=\frac{a^2+1}{a+1} $$ The task is to find ALL possible pairs of $p,q$ for this equation. I've already rewritten that…
Jack09
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Method for Expressing an Integer as the Sum of Three Cubes

I was watching a Numberphile video and I saw how Mathematicians were able to express almost any number as the sum of three cubes$$\begin{align*}1 & =1^3+0^3+0^3\\ & =9^3+10^3+(-12)^3\\29 & =3^3+1^3+1^3\\53 & =27^3+27^3+(-1)^3\\51 &…
Frank
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2-variable Quartic Diophantine equation about number sequences

How could I prove that $$7n^4 + (12s + 6)n^3 + (6s^2 + 6s + 1)n^2 - 2s^3 = 0$$ has no natural solutions? The solutions lead to summation identities and solving this one would prove there are no cubic sequences. On the other hand, any proof that…
sperners lemma
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An identity form: $x_1^3+x_2^3+x_3^3+x_4^3=y_1^3+y_2^3+y_3^3+y_4^3$

I found an identity form: $x_1^3+x_2^3+x_3^3+x_4^3=y_1^3+y_2^3+y_3^3+y_4^3$ as follows: $$(a+b+c)^3+a^3+b^3+c^3=(a+b)^3+(b+c)^3+(c+a)^3+(6y)^3$$ Where $abc=36y^3$ Poof of this identity is very simple. But the identity nice. Which reference of the…
Cố Gắng Lên
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