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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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When is $1^5 + 2^5 + \ldots + n^5$ a square?

When is $1^5 + 2^5 + \ldots + n^5$ a square? I found that this happens sometimes: $n=13$ gives $1001^2$, $n=133$ gives $9712992^2$ and $n=1321$ gives $942162299^2$. I feel that the identity$$\displaystyle\sum_{i=1}^n i^5 =…
sperners lemma
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Why does the diophantine equation $x^2+x+1=7^y$ have no integer solutions?

This following Problem is from Pell equation chapters exercise Let $y>3$ positive integer numbers, show that following diophantine equation $$x^2+x+1=7^y\tag{1}$$ has no integer solutions. I tried write the equation $$(2x+1)^2+3=4\cdot 7^y$$ if…
user246384
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Tricky positive diophantine equation

Find all positive integer solutions to the equation $x^3=y^5+100$. Notice that $7^3=3^5+100$ is a solution, but I can't find any more. Thanks for your help! P.S. I don't know any advanced number theory, just basic olympiad stuff.
Is Ne
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A Diophantine equation solved when N is not a square?

In the following all variables are assumed to be integers. It is easy to write a Diophantine equation which has solutions only when $N$ is a square. i.e. $$N=A^2$$ It's trivial to write a Diophantine equation which has solutions if and only if $N$…
zooby
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Solve this Diophantine equations $(2^x-1)(3^y-1)=2z^2$

Find the positive integers $(2^x-1)(3^y-1)=2z^2$ have three solution $$(1,1,1),(1,2,2),(1,5,11)$$ I already know the solution of $(2^x-1)(3^y-1)=z^2$ has no solution.see:P.G.Walsh December 2006 On Diophantine equations of the form but there is a…
math110
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Positive integer solutions to $x^4+y^7=z^9$

A while ago, a maths teacher gave me this problem: find solutions to $x^4+y^7=z^9$ with $x,y,z>0$. I found $(2^{56})^4+(2^{32})^7=(2^{25})^9$. In general, if $k=8+9l$ then $(2^{7k},2^{4k},2^{\frac{28k+1}{9}})$ is a solution. Then the teacher asked…
Pim
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Another quadratic Diophantine equation: How do I proceed?

How would I find all the fundamental solutions of the Pell-like equation $x^2-10y^2=9$ I've swapped out the original problem from this question for a couple reasons. I already know the solution to this problem, which comes from…
Mike
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Solutions to Diophantine Equation $x^2 - D y^2 = m^2$

I'm looking for solutions to an equation of form $m =\sqrt{x^2 - Dy^2}$. I know that $m$ is a positive integer and so the inside of the square root has to be complete square. So I'm stuck with this diophantine equation $x^2 - Dy^2 = m^2$. In the…
McTrafik
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Is it possible to solve for two unknowns from one equation?

Is it possible to solve for two unknowns using only one equation? For example: $x+3y=32$ Where $x$ and $y$ are integers. Thanks :)
BittersweetNostalgia
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How many answers to $|3^x-2^y|=5$?

How many answers are there to the equation $|3^x-2^y|=5$ such that $x$ and $y$ are positive integers? Are there infinite? I've found $(2,2)$, $(3,5)$, and $(1,3)$. It seems to explode with larger values, but it's not a steady increase and there…
Elem-Teach-w-Bach-n-Math-Ed
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Solving $n! + 3n = k^2$

Let $n$ and $k$ be integers. Need to solve $n! + 3n = k^2$, where $n!$ denotes $n$ factorial. I do not have any ideas about this equation, except I suppose the only $6$ roots are $(0,1), (0, -1), (1, 2), (1, -2), (4, 6),$ and $(4, -6)$ Can anybody…
Dmitry
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Finding a Pythagorean triple $a^2 + b^2 = c^2$ with $a+b+c=40$

Let's say you're asked to find a Pythagorean triple $a^2 + b^2 = c^2$ such that $a + b + c = 40$. The catch is that the question is asked at a job interview, and you weren't expecting questions about Pythagorean triples. It is trivial to look up the…
NPE
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Find all positive integers $L$, $M$, $N$ such that $L^2 + M^2 = \sqrt{ N^2 +21}$

Sorry, this is very much 'can you do my homework' but I have a little competition at work that requires me to solve (and prove) the following. Find all positive integers $L$, $M$, $N$ such that $L^2 + M^2 = \sqrt{ N^2 +21}$. The answers I have are…
brad
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System of Diophantine Equations

I'm working on this problem I came across on the internet but I have no solution yet. The problem states: Find all prime numbers p that are such that $p+1=2x^2$ and $p^2+1=2y^2$ where x and y are integers. Edit: to which point I have reached. I…
user92596
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Positive integral solutions of $3^x+4^y=5^z$

Are there more integral solutions for $3^x+4^y=5^z$, than $x=y=z=2$ ? If not, how do I show that? I could show that for $3^x+4^x=5^x$, but I'm stuck at the general case? Any ideas, maybe graphs, logarithms or infinite descent?
Shubham
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