Questions tagged [diophantine-equations]

Use for questions about finding integer or rational solutions to polynomial equations.

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342
votes
20answers
39k views

Find five positive integers whose reciprocals sum to $1$

Find a positive integer solution $(x,y,z,a,b)$ for which $$\frac{1}{x}+ \frac{1}{y} + \frac{1}{z} + \frac{1}{a} + \frac{1}{b} = 1\;.$$ Is your answer the only solution? If so, show why. I was ...
141
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1answer
4k views

Pythagorean triples that “survive” Euler's totient function

Suppose you have three positive integers $a, b, c$ that form a Pythagorean triple: \begin{equation} a^2 + b^2 = c^2. \tag{1}\label{1} \end{equation} Additionally, suppose that when you apply Euler's ...
110
votes
4answers
126k views

How to find solutions of linear Diophantine ax + by = c?

I want to find a set of integer solutions of Diophantine equation: $ax + by = c$, and apparently $\gcd(a,b)|c$. Then by what formula can I use to find $x$ and $y$ ? I tried to play around with it: $...
84
votes
6answers
18k views

$x^y = y^x$ for integers $x$ and $y$

We know that $2^4 = 4^2$ and $(-2)^{-4} = (-4)^{-2}$. Is there another pair of integers $x, y$ ($x\neq y$) which satisfies the equality $x^y = y^x$?
84
votes
4answers
3k views

How to solve these two simultaneous “divisibilities” : $n+1\mid m^2+1$ and $m+1\mid n^2+1$

Is it possible to find all integers $m>0$ and $n>0$ such that $n+1\mid m^2+1$ and $m+1\,|\,n^2+1$ ? I succeed to prove there is an infinite number of solutions, but I cannot progress anymore. ...
62
votes
0answers
1k views

Finding primes so that $x^p+y^p=z^p$ is unsolvable in the p-adic units

On my number theory exam yesterday, we had the following interesting problem related to Fermat's last theorem: Suppose $p>2$ is a prime. Show that $x^p+y^p=z^p$ has a solution in $\mathbb{Z}_p^{\...
61
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2answers
2k views

On Ramanujan's curious equality for $\sqrt{2\,(1-3^{-2})(1-7^{-2})(1-11^{-2})\cdots} $

In Ramanujan's Notebooks, Vol IV, p.20, there is the rather curious relation for primes of form $4n-1$, $$\sqrt{2\,\Big(1-\frac{1}{3^2}\Big) \Big(1-\frac{1}{7^2}\Big)\Big(1-\frac{1}{11^2}\Big)\Big(1-\...
55
votes
4answers
2k views

Finding triplets $(a,b,c)$ such that $\sqrt{abc}\in\mathbb N$ divides $(a-1)(b-1)(c-1)$

When I was playing with numbers, I found that there are many triplets of three positive integers $(a,b,c)$ such that $\color{red}{2\le} a\le b\le c$ $\sqrt{abc}\in\mathbb N$ $\sqrt{abc}$ divides $(a-...
53
votes
5answers
2k views

Golden Number Theory

The Gaussian $\mathbb{Z}[i]$ and Eisenstein $\mathbb{Z}[\omega]$ integers have been used to solve some diophantine equations. I have never seen any examples of the golden integers $\mathbb{Z}[\varphi]$...
51
votes
3answers
6k views

$n!+1$ being a perfect square

One observes that \begin{equation*} 4!+1 =25=5^{2},~5!+1=121=11^{2} \end{equation*} is a perfect square. Similarly for $n=7$ also we see that $n!+1$ is a perfect square. So one can ask the truth of ...
51
votes
6answers
2k views

Generalizing the sum of consecutive cubes $\sum_{k=1}^n k^3 = \Big(\sum_{k=1}^n k\Big)^2$ to other odd powers

We have, $$\sum_{k=1}^n k^3 = \Big(\sum_{k=1}^n k\Big)^2$$ $$2\sum_{k=1}^n k^5 = -\Big(\sum_{k=1}^n k\Big)^2+3\Big(\sum_{k=1}^n k^2\Big)^2$$ $$2\sum_{k=1}^n k^7 = \Big(\sum_{k=1}^n k\Big)^2-3\Big(\...
50
votes
2answers
2k views

Decomposing polynomials with integer coefficients

Can every quadratic with integer coefficients be written as a sum of two polynomials with integer roots? (Any constant $k \in \mathbb{Z}$, including $0$, is also allowed as a term for simplicity's ...
49
votes
4answers
2k views

Rational solutions to $a+b+c=abc=6$

The following appeared in the problems section of the March 2015 issue of the American Mathematical Monthly. Show that there are infinitely many rational triples $(a, b, c)$ such that $a + b + ...
46
votes
6answers
17k views

Is $\sqrt[3]{p+q\sqrt{3}}+\sqrt[3]{p-q\sqrt{3}}=n$, $(p,q,n)\in\mathbb{N} ^3$ solvable?

In this recent answer to this question by Eesu, Vladimir Reshetnikov proved that $$ \begin{equation} \left( 26+15\sqrt{3}\right) ^{1/3}+\left( 26-15\sqrt{3}\right) ^{1/3}=4.\tag{1} \end{equation} $$ ...
44
votes
3answers
2k views

Triangular Factorials

I came across a statement online and have been looking for a proof : It states that 1, 6 and 120 are the only numbers which are both triangular and factorials. Is there any way I can prove this? ...
42
votes
3answers
2k views

Solutions to $\binom{n}{5} = 2 \binom{m}{5}$

In Finite Mathematics by Lial et al. (10th ed.), problem 8.3.34 says: On National Public Radio, the Weekend Edition program posed the following probability problem: Given a certain number of ...
42
votes
2answers
810 views

Is $n(n+1)$ ever a factorial?

Brocard's problem asks if $(n-1)(n+1)$ is ever a factorial. My question is similar: is $n(n+1)$ ever a factorial? This can be seen as the special case $k=2$ of the question: for $2\le k\le n-2,$ when ...
40
votes
5answers
11k views

Fermat's Last Theorem near misses?

I've recently seen a video of Numberphille channel on Youtube about Fermat's Last Theorem. It talks about how there is a given "solution" for the Fermat's Last Theorem for $n>2$ in the animated ...
38
votes
2answers
2k views

Any positive integer solutions to $x^6+y^{10}=z^{15}$?

This question might be easy. The hard question is this: prove that if $a,b,c\geq3$ then there are no solutions in positive integers $x,y,z$ to $x^a+y^b=z^c$ with $x,y,z$ coprime. This implies Fermat, ...
37
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1answer
2k views

Does an elementary solution exist to $x^2+1=y^3$?

Prove that there are no positive integer solutions to $$x^2+1=y^3$$ This problem is easy if you apply Catalans conjecture and still doable talking about Gaussian integers and UFD's. However, can this ...
36
votes
2answers
1k views

Does $2x^2-1=y^5$ have a solution in integers, with $y>1$?

In my solution to this MSE problem, I noted that $2x^2-1=y^5$ is unlikely to have solutions in integers with $y>1$. Recently, I've tried to find a proof, without success. Following Thomas Andrews'...
35
votes
4answers
2k views

Conjecture: Only one Fibonacci number is the sum of two cubes

As the title says, I need help proving or disproving that there is only one Fibonacci number that's the sum of two (positive) cubes, $2$. I did a small brute force test with Fibonacci numbers below $...
35
votes
1answer
4k views

How to compute rational or integer points on elliptic curves

This is an attempt to get someone to write a canonical answer, as discussed in this meta thread. We often have people come to us asking for solutions to a diophantine equation which, after some clever ...
34
votes
9answers
6k views

Linear diophantine equation $100x - 23y = -19$

I need help with this equation: $$100x - 23y = -19.$$ When I plug this into Wolfram|Alpha, one of the integer solutions is $x = 23n + 12$ where $n$ is a subset of all the integers, but I can't seem ...
33
votes
3answers
2k views

Does the equation $x^4+y^4+1 = z^2$ have a non-trivial solution?

The background of this question is this: Fermat proved that the equation, $$x^4+y^4 = z^2$$ has no solution in the positive integers. If we consider the near-miss, $$x^4+y^4-1 = z^2$$ then this has ...
32
votes
2answers
3k views

Diophantine applications of Spec?

Let $f(\bar x)$ be a multivariable polynomial with integer coefficients. The zeros of that polynomial are in bijection with the homomorphisms $\mathbb Z[\bar x] \rightarrow \mathbb Z$ that factor ...
32
votes
2answers
26k views

The equation $x^3 + y^3 = z^3$ has no integer solutions - A short proof

Can someone provide the proof of the special case of Fermat's Last Theorem for $n=3$, i.e., that $$ x^3 + y^3 = z^3, $$ has no positive integer solutions, as briefly as possible? I have seen some ...
32
votes
2answers
1k views

For what kind of positive integers $m+n+m^2+n^2 \mid m^3+n^3$ holds?

Determine all pairs of positive integers $(m, n)$ such that$$\frac{m^3+n^3}{m^2+n^2+m+n}$$is an integer. This is a simpler version of a problem that I made a few years ago. There are infinitely many ...
31
votes
3answers
1k views

Trigonometric diophantine equation $8\sin^2\left(\frac{(k+1)\pi}{n}\right)=n\sin\left(\frac{2\pi}{n}\right)$

Background. I thought up the problem of finding a regular $n$-sided polygon that has a diagonal with length $d_k$ such that the area of the polygon equals ${d_k}^2$. (Let $d_k$ denote the length of a ...
30
votes
6answers
6k views

What is the smallest integer greater than 1 such that $\frac12$ of it is a perfect square and $\frac15$ of it is a perfect fifth power?

What is the smallest integer greater than 1 such that $\frac12$ of it is a perfect square and $\frac15$ of it is a perfect fifth power? I have tried multiplying every perfect square (up to 400 by two ...
30
votes
5answers
842 views

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$$\...
29
votes
3answers
8k views

Find integer in the form: $\frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b}$

Let $a,b,c \in \mathbb N$ find integer in the form: $$I=\frac{a}{b+c} + \frac{b}{c+a} + \frac{c} {a+b}$$ Using Nesbitt's inequality: $I \ge \frac 32$ I am trying to prove $I \le 2$ to implies there ...
29
votes
2answers
580 views

Find $a,b,c,d,e$ such that $\frac{s}a+1,\frac{s}b+1,\frac{s}c+1,\frac{s}d+1,\frac{s}e+1$ are all perfect squares $ (s=abcde)$

Are there five distinct positive integers $a,b,c,d,e$ such that $\dfrac{s}a+1,\dfrac{s}b+1,\dfrac{s}c+1,\dfrac{s}d+1,\dfrac{s}e+1$ are all perfect squares ? $ (s=abcde)$ If $a=1,b=2,c=12,d=2380,s=...
28
votes
6answers
868 views

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 ...
28
votes
3answers
575 views

On the arithmetic differential equation $n''=n'$

If $n'$ denotes the arithmetic derivative of non-negative integer $n$, and $n''=(n')'$, then solve the following equation $$n''=n'.$$ What I have found, you can read in one minute! I have tried to ...
28
votes
1answer
635 views

For integers $a\ge b\ge 2$, is $f(a,b) = a^b + b^a$ injective?

Given two integers $a \ge b \ge 2$, can we encode them as a unique integer $a^b + b^a$? This question was asked a few weeks ago, but did not rule out the trivial cases. For example, if we allow one ...
27
votes
8answers
4k views

Variation of Pythagorean triplets: $x^2+y^2 = z^3$

I need to prove that the equation $x^2 + y^2 = z^3$ has infinitely many solutions for positive $x, y$ and $z$. I got to as far as $4^3 = 8^2$ but that seems to be of no help. Can some one help me ...
27
votes
5answers
11k views

Can n! be a perfect square when n is an integer greater than 1?

Can n! be a perfect square when n is an integer greater than 1? (But is it possible, to prove without Bertrand's postulate. Because bertrands postulate is quite a strong result.)
27
votes
8answers
2k views

Examples of Diophantine equations with a large finite number of solutions

I wonder, if there are examples of Diophantine equations (or systems of such equations) with integer coefficients fitting on a few lines that have been proven to have a finite, but really huge number ...
27
votes
2answers
658 views

A diophantine equation with only “titanic” solutions

I made a note some time ago that I had read in some book that the equation $$313(x^3+y^3)=t^3$$ has positive integer solutions, but that these are so large that it would be absolutely hopeless to ...
26
votes
2answers
1k views

If $p$, $q$ are naturals, solve $p^3-q^5=(p+q)^2$.

In If $p,q$ are prime, solve $p^3-q^5=(p+q)^2$., the author asks to solve the equation $p^3-q^5=(p+q)^2$ for primes $p$ and $q$. A proof is given that $p=7, q=3$ is the only solution. In this "...
26
votes
1answer
389 views

Numbers that are clearly NOT a Square

Although I have never studied math very seriously, I have heard of Brocard's Problem, which asks for integer solutions for the following Diophantine Equation:$$n!+1=m^2$$ The only solutions are ...
25
votes
4answers
1k views

Minimum of $n$? $123456789x^2 - 987654321y^2 =n$ ($x$,$y$ and $n$ are positive integers)

What is the minimum of $n$? $x$,$y$ and $n$ are positive integers, find the minimum of $n$, such that: $123456789x^2 - 987654321y^2 =n$
24
votes
2answers
2k views

Equation with solution in prime numbers

How many solutions are there for the equation $x^y-y^x=x+y$ where $x$ and $y$ are prime? I have found one $2^5-5^2=2+5$ and I'm not sure whether it's the only solution..
23
votes
3answers
3k views

Find a thousand natural numbers such that their sum equals their product

The question is to find a thousand natural numbers such that their sum equals their product. Here's my approach : I worked on this question for lesser cases : \begin{align} &2 \times 2 = 2 + 2\\ ...
23
votes
9answers
12k views

Diophantine equation $a^2+b^2=c^2+d^2$

I was reasonably certain I've seen this before, but I was wondering how to solve the Diophantine equation $$a^2+b^2=c^2+d^2$$ I tried a web search and found nothing on this one. I'm trying to avoid ...
23
votes
0answers
975 views

Are there unique solutions for $n=\sum_{j=1}^{g(k)} a_j^k$?

Edward Waring, asks whether for every natural number $n$ there exists an associated positive integer s such that every natural number is the sum of at most $s$ $k$th powers of natural numbers (for ...
22
votes
6answers
2k views

A very different property of primitive Pythagorean triplets: Can number be in more than two of them?

While playing with numbers, I thought about squares of numbers, and then the first thing that came to mind was Pythagorean triplets. I observed a very interesting fact that any $x\in\mathbb N$ can ...
22
votes
8answers
7k views

Number of integral solutions for $|x | + | y | + | z | = 10$

How can I find the number of integral solution to the equation $|x | + | y | + | z | = 10.$ I am using the formula, Number of integral solutions for $|x| +|y| +|z| = p$ is $(4P^2) +2 $, So the ...
22
votes
7answers
1k views

Integer solutions to $x^3=y^3+2y+1$?

Find all integral pairs $(x,y)$ satisfying $$ x^3=y^3+2y+1.$$ My approach: I tried to factorize $x^3-y^3$ as $$(x-y)(x^2 + xy + y^2)=2y+1,$$ but I know this is completely helpless. Please help me in ...

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