Questions on finding integer/rational solutions of polynomial equations.

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-2
votes
1answer
41 views

Solve for b and d

Solve for b and d in the following equation. A triangle with sides $(a, a, b)$ has the same area and the same perimeter as a triangle with sides $(c, c, d)$ where $a, b, c$ and $d$ are positive ...
-1
votes
0answers
23 views

Diophantine Equation with gcd. [duplicate]

Find all positive integers $a,b$ such that $\gcd(3^a+1,3^b+1)$ is a multiple of $ab$. I've given this problem many attempts but I can't seem to make any progress, there doesn't appear to be any way ...
4
votes
1answer
63 views

Substitutions that transform Fermat Equations to Elliptic Curves

I was reading Chapter 1 of Elliptic Curves - Number Theory and Cryptography by Lawrence C Washington. He was considering Fermat equations $$a^4+b^4=c^4\text{ and }a^3+b^3=c^3.$$ For the 1st equation, ...
3
votes
2answers
111 views

Showing there is no triplet of positive integers $(a,b,c)$ satisfying $a^7+b^7=7^c$ [duplicate]

Show that $$a^7+b^7=7^c$$ has no positive integer solutions $(a,b,c)$. I've posted a general and way too long approach as an answer. How may one prove the claim more briefly and specifically?
4
votes
3answers
203 views

Solving $x^p + y^p = p^z$ in positive integers $x,y,z$ and a prime $p$

The question is from Zeitz's ''The Art and Craft of Problem Solving:" Find all positive integer solutions $x,y,z,p$, with $p$ a prime, of the equation $x^p + y^p = p^z$. One thing I noticed is ...
2
votes
1answer
145 views

Solve $x^n+z^n=(x+1)^n$ for $n\ge 3$ without FLT

Is there a way to prove that for $x,z,n \in \mathbb{Z}$, $x > 0$, $z > 0$, $n > 2$, the equation $$ x ^ n + z ^ n = (x + 1) ^ n $$ has no solution, without using Fermat's Last Theorem? ...
0
votes
1answer
61 views

Can the solvability of a single Diophantine equation be undecidable (in any sense of the word)?

Apologies in advance for asking the following "philosophical" question, which falls dramatically short of any reasonable standards of mathematical rigour: Is it possible that there should exist a ...
0
votes
1answer
67 views

Are all even numbers the difference of prime powers

Does there exist an even positive integer greater than $100$ (to eliminate trivial cases) that cannot be expressed in the form: $p^2-q$ $p-q^2$ $p^2-q^2$ $p^3-q^3$ where $p$ and $q$ are primes.
237
votes
13answers
29k 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 ...
4
votes
2answers
88 views

$2^x+7^y=19^z$ has no solution in positive integers $x$, $y$, $z$

How do I show that the diophantine equation $2^x+7^y=19^z$ has no solution in positive integers $x$, $y$, $z$
4
votes
1answer
432 views

On the Cubic Pell equation $x^3+dy^3+d^2z^3-3dxyz =1$ for $d = 23$ etc.

(This post was motivated by an old one.) For Pell equations, $$x^2-dy^2 = 1\tag{1}$$ and $d<100$, the largest fundamental solution is for $d = 61$ (which happens to be the 6th power of a ...
0
votes
1answer
37 views

Postive integer solution to this equation $a^2+b^2+c^2+1=kabc$

Frobenius and Hurwitz( in 1880) prove this theorem: For any positive integer $k$ other than 1 or 3, the equation $a^2+b^2+c^2=kabc$ has no integral solution except (0,0,0). My Question,How to solve ...
5
votes
1answer
61 views

How to extract solutions to a Pell's equation satisfying certain congruences?

I'm trying to solve $y^2=3x^2+3x+1$ for integers, which transforms into $(2y)^2-3(2x+1)^2=1$. I know how to solve pell's equation, but how can we extract only (odd,even) pair from the solutions of the ...
1
vote
1answer
52 views

How to solve a bivariate quadratic (not necessarily Pell-type) equation?

Simple Pell equations often have solutions that can be found with little work given certain conditions. These are of the form $x_{n}^{2} - A y_{n}^{2} = \pm 1$. There are harder equations that involve ...
7
votes
5answers
246 views

Which triangular numbers are also squares?

I'm reading Stopple's A Primer of Analytic Number Theory: Exercise 1.1.3: Which triangular numbers are also squares? That is, what conditions on $m$ and $n$ will guarantee that $t_n=s_m$? Show ...
6
votes
1answer
135 views

To solve $n(n+1)(n+2)=6m^3$ in positive integers $m,n$

How to find all positive integers $m,n$ such that $n(n+1)(n+2)=6m^3$ ? I can see that $m=n=1$ is a solution , but is it the only solution ?
1
vote
2answers
50 views

Square numbers in the form $1+4y$

I want to solve the equation $y+x=x^2$: $$ x^2-x-y=0 \\ x_{1;2}=\frac{1\pm \sqrt{1+4y}}{2} $$ However I want the solutions to be only natural numbers; the question then turns to find values of $y$ ...
3
votes
1answer
96 views

To find positive integers $n$ such that $\dfrac {n(n+1)(n+2)}6$ is a perfect square [duplicate]

How many positive integers $n$ are there such that $\dfrac {n(n+1)(n+2)}6$ is a perfect square ? I know $n=1 , 2$ works ; are there any more ? Are there only finitely many such $n$ ?
2
votes
3answers
133 views

Markov-Hurwitz equation

Prove that the Markov-Hurwitz equation $x^2+y^2+z^2=dxyz$ is solvable in positive integers iff d= 1 or 3. Of course the reverse direction is easy, just set x=y=z=1, d=3. But I really have no idea ...
4
votes
1answer
126 views

Integer solutions of $a^3+2a+1=2^b$

What are the solutions in integers of $a^3+2a+1=2^b$? [Source: Serbian competition problem]
1
vote
2answers
44 views

Concluding three statements regarding $3$ real numbers.

$\{a,b,c\}\in \mathbb{R},\ a<b<c,\ a+b+c=6 ,\ ab+bc+ac=9$ Conclusion $I.)\ 1<b<3$ Conclusion $II.)\ 2<a<3$ Conclusion $III.)\ 0<c<1$ Options By ...
4
votes
2answers
126 views

Diophantine Equation $ x^n + y^n =z^n (x<y, n>2) $

I am looking for simple college level algebraic solution to prove that $x$ and $y$ ($x$ < $y$) for the above equation can't be prime numbers. (I know more complex and involved solution using high ...
1
vote
2answers
49 views

Write $x(a^2+b^2)+(2ab)y$ as a product of factors.

Let $a,b,c,x,y \in\mathbb{Z}>1$ and $\gcd(a,b)=\gcd(x,y)=\gcd(a,b,x,y)=1$, Can $$x(a^2+b^2)+(2ab)y$$be factorized ?
9
votes
0answers
82 views

Generalizing the Pell equation $x^2-61y^2 = 1$

In a table of fundamental solutions $f_1(x,y)$ to Pell equations, $$x^2-dy^2=1\tag1$$ with $d<110$, two will stand out, $$(U_{61})^6 = \big(\tfrac{39+5\sqrt{61}}{2}\big)^6 = x+y\sqrt{61} ...
14
votes
5answers
2k views
1
vote
3answers
76 views

Maximize the following sum

Let $a, b, c, d, e$ be nonnegative integers such that $625a + 250b + 100c + 40d + 16e = 15^3$ . What is the maximum possible value of $a + b + c + d + e$? Quick arithmetic gives: ...
1
vote
1answer
25 views

for which values of $\theta$.does this equation: $x_1^{\sin\theta}+\cdots+x_n^{\sin\theta}=1$ have rational solutions for all $n$?

I'd surprised if this equation:$$x_1^{\sin\theta}+\cdots+x_n^{\sin\theta}=1 $$ have rational solutions for all $n$ and for all values of $\theta$. My question here is: for which values of $\theta$ ...
0
votes
2answers
35 views

How many non-negative integer solutions does $x_1+x_2+\cdots+x_n=A$ have?

If I have the Diophantine equation $\displaystyle{\sum_{i=1}^n x_i =A}$, is there a function $f(n,A)$ that will yield the number of non-negative integer solutions of the equation?
2
votes
3answers
85 views

Almost extended Euclidean algorithm - $ax+by=\gcd(a,b)+2$

So I have this equation: $$\eta+2=2g+1n,$$ where $g,n \in \mathbb{N}_{\geq 0}$ and $\eta \in \mathbb{N}_{>0}$. I want to find all possible integer-valued 2-tuples $(g,n)$ that satisfy this ...
4
votes
3answers
129 views

Solve in positive integers: $5^x 7^y +4=3^z$

Solve in positive integers: $5^x 7^y +4=3^z$. I tried to solve it with log but I couldn't complete.
0
votes
0answers
38 views

The amount of the third degree.

Often have to deal with such a cubic Diophantine equation. $$q(a^3+b^3)=t(x^3+y^3)$$ $q,t - $ are specified for the problem. Interesting - in all the values of the coefficient of solutions are ...
1
vote
5answers
142 views

Finding integers of the form $m+n$ that satisfies $m+n+mn=118$

Let $m$ and $n$ be two positive integers such that $m+n+mn=118.$ My question is: Can the value of $(m+n)$ be uniquely determined? I find by inspection that the pair $(m,n)=(16,6)$ (or the ...
20
votes
6answers
2k views

Chicken Problem from Terry Tao's blog (system of Diophantine equations)

This problem was posted by Terry Tao in his blog earlier. It's actually from his son's Math Circle. It took him $15$ minutes to solve it. I guess we all can take a crack at it. Three farmers were ...
4
votes
1answer
36 views

Solving a Diophantine equation with LTE

Show that only positive integer value of $a$ for which $$4(a^n+1)$$ is a perfect cube for all positive integers $n$, is $1$. Rewriting the equation we obtain: $$4(a^n+1)=k^3$$ It is obvious that $k$ ...
2
votes
2answers
63 views

Are there any primes for which $a^2 = pb^2 + 1$ does not exist?

The smallest solution to the above equation for various primes are: $(p=2)$ $3^2 = 2*2^2 +1$ $(p=3)$ $2^2 = 2*1^2 +1$ $(p=5)$ $9^2 = 5*4^2 +1$ $(p=7)$ $8^2 = 7*3^2 +1$ Is there at least one ...
2
votes
4answers
278 views

System of quadratic Diophantine equations

Is there a method for determining if a system of quadratic diophantine equations has any solutions? My specific example (which comes from this question) is: $$\frac{4}{3}x^2 + \frac{4}{3}x + 1 = ...
-1
votes
3answers
88 views

Integral Solutions of $x+y=x^2-xy+y^2$

Find all integral solutions of $$x+y=x^2-xy+y^2$$ A modulo 2 analysis does not work here, only says cannot both be odd.
0
votes
1answer
33 views

Solve in positive integers $a^2-b^2+4a=0$

Solve in positive integers $$a^2-b^2+4a=0$$ I tried considering the residues in mod4 but not so helpful. Any help/hint on how to approach this problem ? Thanks !
0
votes
2answers
97 views

Find all natural number solutions to: $20x^2 + 11y^2 = 2011$

I believe that the equation $$20x^2 + 11y^2 = 2011$$ describes an ellipse. I don't know how to solve for the $x,y \in \mathbb{N}$ that satisfy this equation.
3
votes
3answers
112 views

Find all Integral solutions to $x+y+z=3$, $x^3+y^3+z^3=3$.

Suppose that $x^3+y^3+z^3=3$ and $x+y+z=3$. What are all integral solutions of this equation? I can only find $x=y=z=1$.
0
votes
0answers
41 views

For which values of $k$ does: $ y^2 = x^3+(2^{2^k}+1)x$ have solutions in integers?

let $E_D$ be elliptic curve and $k$ is integer number $$E_D: y^2 = x^3+px. $$ When $p = 2^{2^k}+1$ is prime fermat . my question is :For which values of $k$ does:E.d $$ y^2 = x^3+px. $$ have ...
3
votes
1answer
23 views

Hensel lifting when not a power of a prime

Say you have the equation $x^2 + x + 47 = 0$ and that you want to determine the solutions in $\mathbb{Z}/1715 \mathbb{Z}$. Note that $1715 = 7^3 \cdot 5$. Then, using Hensel's lemma, one can find the ...
3
votes
2answers
61 views

Show that there are infinitely many integer solutions to the equation $x^3+y^5=z^7$

Show that there are infinitely many integers such that $$x^3+y^5=z^7$$ and where $x^3,y^5$ and $z^7$ are all non-zero and distinct. The hint suggests to look at solutions of simultaneous equation ...
4
votes
0answers
159 views

Solve in integers $ (y^3+xy-1)(x^2+x-y)=(x^3-xy+1)(y^2+x-y)$

Solve in integers: $$ (y^3+xy-1)(x^2+x-y)=(x^3-xy+1)(y^2+x-y)$$ My idea: $$\Longleftrightarrow (y^3+xy-1)(x^2+x-y)-(x^3-xy+1)(y^2+x-y)=0$$ $$\Longleftrightarrow ...
1
vote
0answers
42 views
12
votes
2answers
546 views

Diophantine equation involving prime numbers : $p^3 - q^5 = (p+q)^2$

Find all pairs of prime nummbers $p,q$ such that $p^3 - q^5 = (p+q)^2$. It's obvious that $p>q$ and $q=2$ doesn't work, then both $p,q$ are odd. Assuming $p = q + 2k$ we conclude, by the equation, ...
9
votes
1answer
115 views

Integer solutions to the equation $a_1^2+\cdots +a_n^2=a_1\cdots a_n$

What is the general solution to the equation $$\sum_{j=1}^n a_j^2=\prod_{j=1}^n a_j,$$ $n\in \mathbb N$ , $n \ge 2$ over $\mathbb N_0$ ? WLOG, we can assume $0\le a_1 \le a_2\le \cdots \le a_n$ For ...
3
votes
3answers
652 views

Primitive integer triangles

Consider the triangles with integer sides $a$, $b$ and $c$ with $a \leq b \leq c$. An integer sided triangle $(a,b,c)$ is called primitive if $gcd(a,b,c)=1$. How many primitive integer sided triangles ...
2
votes
2answers
128 views

Why is ${n\choose k}$ is always a product of the primes of $n$ for all $n>k$? [closed]

Let $n, k$ be two positive integers such that $n>k$. Why is ${n\choose k}$ always divisible by a prime dividing $n$ (or even a product of such primes)? Please help me understand why. I cannot seem ...
3
votes
1answer
37 views

Fruit vendor selling fruit

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with simultaneous diophantine equations, but other than that, the textbook gave ...