Questions on finding integer/rational solutions of equations.

learn more… | top users | synonyms

11
votes
0answers
556 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 ...
10
votes
0answers
211 views

Help with $x^4+y^4+z^4 = 1$?

There are exactly 20 known primitive solutions to, $$a^4+b^4+c^4 = d^4\tag{1}$$ with $d<10^{10}$. Noam Elkies (who kindly answered Question 1 below) showed that the form, $$(x+y)^4+(x-y)^4+z^4 = ...
10
votes
0answers
215 views

Primes as quotients

I ask this question based on a comment of David Speyer in another question. What primes are of the form $$ \frac{p^2-1}{q^2-1} $$ where $p$ and $q$ are prime? The first prime not apparently of this ...
8
votes
0answers
128 views

Solve the Diophantine equation $a^2(2^a-a^3)+1=7^b$.

The problem is to find all positive integers $a$ and $b$ such that $a^2(2^a-a^3)+1=7^b$. I found a=10, and my intuition tells me there are no more solutions. I've also shown that $a=42k+10$ for some ...
8
votes
0answers
544 views

How many integer solutions to a diophantine equation

Starting with the equation: $\frac{1}{a}+\frac{1}{b}=\frac{p}{10^n}$, I reached the equation: $10^{n-log(p)} = \frac{ab}{a+b}$. Now given the positive integer $n$, for what integer values of $p$ ...
7
votes
0answers
182 views

Counting the Number of Integral Solutions to $x^2+dy^2 = n$

It is a well known result that the number of integer solutions $(x,y), x>0, y\ge 0$ to $x^2+y^2 = n$ is $\sum_{d|n}\chi(d)$, where $\chi$ is the nontrivial Dirichlet character modulo $4$ such that ...
7
votes
0answers
103 views

Geometric intuition behind the Hasse principle

Let $f(X,Y) \in \mathbb{Q}[X,Y]$ be a quadratic polynomial. The Hasse-Minkowski theorem says that $f(X,Y) = 0$ has a solution $(x,y) \in \mathbb{Q}^2$ iff it has a solution in $\mathbb{R}^2$ and ...
6
votes
0answers
141 views

Coefficients in expansion of $(\sqrt[3]{2} - 1)^m$

In trying to solve $a^3 - 2b^3 = 1$ over the integers I came across the need to answer the question: when does $(1+ \sqrt[3]{2} + \sqrt[3]{2}^2)^n$ have no $\sqrt[3]{2}^2$ term in it's expansion (in ...
6
votes
0answers
108 views

Twin Prime Powers

What are all the possible triplets of numbers $a$, $b$, $c$ such that $a+2=b$, $a+4=c$, and all $3$ are prime powers (where one must be a power of $3$)? I'm aware of the cases for when they are ...
6
votes
0answers
205 views

Diophantine special problem

This is my another question on Diophantine equations. Prove the following great and special problem. Let $D$ and $k$ be positive integers and $p$ be a prime number such that $gcd(D, kp) = 1$. Prove ...
6
votes
0answers
311 views

On the equation $m^3-m^2+1 = n^2$

(i) How can I find all positive integers $m$ such that $m\equiv 4 \pmod 7$ and $m^3-m^2+1$ is a perfect square? (ii) Is there a method to solve this equation over positive integers: $$m^3-m^2+1 = ...
5
votes
0answers
79 views

About solutions to $x^2+y^2+z^2=8n+3,n\in \mathbb N$

As we know that $x^2+y^2+z^2=8n+3,(n\in \mathbb N)\tag{1}$ has integer solutions $x,y,z\in \mathbb N.$ If $k\in \mathbb N$ has at least one prime factor which is $\equiv 3 \mod 4,$ then we call $k$ ...
5
votes
0answers
171 views

Ramanujan-Nagell Theorem Proof Question

I'm currently working through Stewart and Tall's Algebraic Number Theory. In particular, section 4.9 of this book provides a proof of the Ramanujan-Nagell Theorem, which states that the only integer ...
5
votes
0answers
100 views

Is there always a telescopic series associated with a rational number?

Here is something I thought up while I was bored and my, erm, fish were busy: Given a rational number $p\in(0,1)$, are there always positive integers $n$, $c_m$, and $w_m$ such that ...
4
votes
0answers
83 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 $x^3-y^5=(x+y)^2$ for primes $p$ and $q$. A proof is given that $p=7, q=3$ is the only solution. In this ...
4
votes
0answers
87 views

Generalizing Ramanujan's 6-10-8 Identity

Let $ad=bc$. Then Ramanujan's 6-10-8 Identity is the bizarre, $$64[(a+b+c)^6+(b+c+d)^6-(c+d+a)^6-(d+a+b)^6+(a-d)^6-(b-c)^6][(a+b+c)^{10}+(b+c+d)^{10}-(c+d+a)^{10}-(d+a+b)^{10}+(a-d)^{10}-(b-c)^{10}] ...
4
votes
0answers
153 views

How many natural numbers $x, y$ are possible if $(x - y)^2 = \frac{4xy}{(x + y - 1)}$.

How many natural numbers $x$, $y$ are possible if $(x - y)^2 = \frac{4xy}{x + y - 1}$. Does this system has infinite solutions which can be generalized for some integer $k \geq 2?$ $(x - y)^2(x + y) ...
3
votes
0answers
92 views

diophantine equation $ |x^2-py^2|=\frac{p-1}{2} $

Prime $p\equiv3\pmod4$, then diophantine equation $$ |x^2-py^2|=\frac{p-1}{2} $$ has a solution in integers en, $x^2-py^2=-1$ has no solution in integers. I'd be grateful for any help you are ...
3
votes
0answers
62 views

Find all $x,y\in\mathbb{Z}$ s.t $2x^3-7y^3=3$

Find all $$x,y\in\mathbb{Z}$$ such that $$2x^3-7y^3=3$$ Solution: We consider first $$2x^3-7y^3\equiv3 \pmod 2$$ $$5y^3\equiv 1 \pmod 2$$ $$y^3\equiv 1 \pmod2$$ which has solution $y\equiv 1 ...
3
votes
0answers
94 views

General quadratic diophantine equation.

Here is my problem: I am given a general quadratic diophantine equation: $$ax^2 + bxy + cy^2 + dx + ey + f = 0$$ where $x$ and $y$ are variables with integers $a,b,c,d,e,f$. I have to show that if the ...
3
votes
0answers
64 views

How to solve $x^4+y^4=n$?

How to solve Diophantus equation $$x^4+y^4=n $$ where $x,y$ and $n$ are positive integers. We know that Theorem: A natural numbern $n$ can be represented as a sum of two squares if and only if ...
3
votes
0answers
147 views

Second longest prime diagonal in the Ulam spiral?

Given the Ulam spiral with center $C = 41$ and the numbers in a clockwise direction, we have, $$\begin{array}{cccccc} \color{red}{61}&62&63&64&\to\\ ...
3
votes
0answers
72 views

Solving $(x^m+y^m-z^m)^n=(x^n+y^n-z^n)^m$

If $m$ and $n$ are distinct positive integers then does the equation $(x^m+y^m-z^m)^n=(x^n+y^n-z^n)^m$ $\space$has any solution , for $x,y,z$ , in positive integers with $x,y,z$ all not equal ?
3
votes
0answers
86 views

how many natural numbers on a sphere

how many natural solutions are there to the following equation: $$ \sum_{i=0}^k x_i^2 = n$$ where $n,k \in\ \Bbb{N}$ i well like to get a answer for every n and k, but could do with just $k=2,3$.
3
votes
0answers
235 views

Diophantine with Gaussian Integer

I'm trying to find the set of solutions to a specific diophantine equation over $\mathbb{Z}[i]$. The equation is the following: $$ z_1^2 + z_2^2 + z_1*z_2 + 39 = 0$$ with $ z_1$ (resp $z_2$) such ...
3
votes
0answers
82 views

Do the solutions to the unit equation lie dense in the complex numbers

Let $S\subset \overline{\mathbf{Q}}$ be the set of solutions to the unit equation, i.e., $S$ consists of algebraic integers $a$ such that $a$ and $1-a$ are units in the ring of algebraic integers. ...
3
votes
0answers
349 views

$XX^t=A$, $X=?$. Where $X \in \{0,1\}^{n \times m}$

The problem: $XX^t=A$, $\quad$ ($X_{ij}\in{0,1}$, $\quad$ $\sum_{j=1}^m x_{ij}=2$), $\quad$ $X=?$ Details: $n,m \in N$ $A \in \{0,1,2\}^{n \times n}$ $X \in \{0,1\}^{n \times m}$ $A$ is a ...
3
votes
0answers
246 views

Counting Solutions of Diophantine Inequalities

I understand that Diophantine Analysis is an enormous field! Without first determining the solution set, suppose I'd like to calculate the number of non-negative integer solutions $(x,y,z)$ of ...
2
votes
0answers
48 views

How find this integer $x,y$ such $1+5^x=2\cdot 3^y$

Find this equation integer solution $$1+5^x=2\cdot 3^y$$ I know $$x=1,y=1$$ is such it.and $$x=0,y=0$$ This problme is Shanghai mathematics olympiad question in 2014 I think this equation have no ...
2
votes
0answers
39 views

An algorithm for solving linear diophantine equations?

I am entering an interesting team based math contest called the purple comet, and quite a lot of questions on this contest involve Diophantine equations. For this contest, you are given a computer, ...
2
votes
0answers
57 views

Find all integers n which satisfies $1^n+9^n+10^n=5^n+6^n+11^n$

Find all $n\in\mathbb Z$ which satisfies $1^n+9^n+10^n=5^n+6^n+11^n$ for $n=2\ or\ n=4$ it is equal but are there other numbers?
2
votes
0answers
40 views

Using Graphs Changes the Solutions for Diophantine Equation? Imperfection of Graph?

Solve the Diophantine equation $$x^2+4y^2=z^2$$ The problem here is that I derived solutions using two different methods, and the both solutions do satisfy the given equation yet they are ...
2
votes
0answers
55 views

fourth powers as sums of squares

Is it possible to have a fourth power that is the sum of two squares in four different ways, e.g., $w^4 = a^2 + b^2 = c^2 + d^2 = e^2 + f^2 = g^2 + h^2$ with the added restriction that $e = a+c$ and ...
2
votes
0answers
73 views

Find all Integers ($ n$) such that $n\neq 6xy\pm x\pm y$

I am interested in proving that there exist an infinite number of positive integers ($n$) which are not of the form $$ n=6xy\pm x\pm y $$ for $x,y\in\Bbb Z^+$. [Note: The $\pm$ signs above are ...
2
votes
0answers
45 views

Abelian SubGroup Variant:

Consider the following problem: Find integers $x_1, x_2, x_3,\dots, x_n$ Such that: $$P(x_1,x_2,\dots, x_n) = Q$$ for some integer $Q$ and polynomial $P$ where for all permutations of any set of ...
2
votes
0answers
98 views

Count number of positive integer solutions of $x^2(8x-3)=y^2z$?

Given the Diophantine equation $$ x^2(8x-3)=y^2z, $$ is there a way to efficiently count the number of solutions that satisfy $x+y+z\leq n$, where $n$ is a fixed given integer? Also, for any fixed ...
2
votes
0answers
72 views

Solving a particular system of Diophantine equations in $n$ variables (Frobenius equations)

I have a particular system of linear Diophantine equations in $n$ variables for which I need to find all nonnegative integer solutions. Specifically, they are Frobenius equations, meaning the ...
2
votes
0answers
90 views

Quadratic Diophantine Equations in Polynomial Time

Considering the problem of finding lattice points $(x_1, x_2 ... x_n)$ that satisfy a quadratic law: $F(x_1, x_2... x_n) = 0$ such that $F(x_1, x_2... x_n)$ is a second degree polynomial It is ...
2
votes
0answers
102 views

Diophantus again; not to say Pell.

Is there a way to solve the second degree Diophantine equation in two variables $ax^{2} -ny^{2} = b$ $(1)$ where a and b are known and n is a parameter; all solutions x= f(n) and y = f(n) ? For ...
2
votes
0answers
144 views

Diophantine equations/Diophantine Geometry

I am very knew to this site and I am eagerly waiting for solutions of: (1) Let $x$ be an algebraic number with degree $n > 1$. Then there exists only finitely many rational numbers $p/q$ (in ...
2
votes
0answers
32 views

Solving $key=(\sum_{K=0}^n\frac{1}{a^K})\mod m$ with High limits

I was solving this equation:- $$key=(\sum_{K=0}^n\frac{1}{a^K})\mod m$$ Given $$ 1,000,000,000 < a, n, m \; < 5,000,000,000 $$ $$ a, m \; are \;coprime $$ I solved it bruteforcely but it ...
2
votes
0answers
184 views

Pell equation proof

Prove that whenever the equation $x^2 - dy^2 = c$ is solvable, then it has infinitely many solutions. I consider that, if $u$ and $v$ satisfy $x^2 -dy^2 = c$ and then $r$ and $s$ satisfy $x^2 ...
2
votes
0answers
399 views

A Quadratic diophantine equation

How to prove or disprove this statement: For all $c<z<0<s$, there exists $0<k\leq i$, $0\leq j<s+i$, such that all conditions hold simultaneously: ...
2
votes
0answers
138 views

Does $~4^y+1=4xy^2+x~$ have infinitely many solutions in integers?

Consider the equation : $~4^y+1=4xy^2+x$ I have found that this equation has integer solutions for following values of $~y$ : $y\in \{1,2,193,10068,29570,..\}$ Question : Are there finitely or ...
2
votes
0answers
174 views

Upper bound for the quality of an $abc$-triple

A triple of positive integers $(a,b,c)$ is an $abc$-triple if $a$ and $b$ are coprime and $c = a + b$. Define the quality or power of an $abc$-triple as $P(a,b,c) = \frac{\log c}{\log ...
1
vote
0answers
46 views

Diophantine Equation $2x^2+25=y^3$

I'm trying to find integer solutions to: $2x^2+25=y^3$. Here's what I've managed to do so far: y is odd. y and x are co-prime. In $\mathbb{Q}(\sqrt{2},i)$ we can write: ...
1
vote
0answers
98 views

15th power Diophantine equation

I'd appreciate some help (a hint) for the following. If $x,y>1$ are so that $2x^2-1=y^{15}$ then $x$ is a multiple of $5$. Don't know if this helps but the equation can be rewritten as ...
1
vote
0answers
55 views

How to solve diophantine equation $\frac{x^p-y^p}{x-y}=n$

$$\frac{x^p-y^p}{x-y}=n$$ whit $p$ a prime greater than or equal to $3$,for what value to $n$, it's solvable and how to solve,and whether $\frac{x^p-y^p}{x-y}=q_1$ $\frac{x^p-y^p}{x-y}=q_2$ is ...
1
vote
0answers
32 views

Particular case of the n-degree equation.

It's know that there isn't a general formula for solve the general n-degree equation for $n>4$, but there is any formula to solve the case? $A^x+A=C$ where $A$ is the variable and $A,C$ are ...
1
vote
0answers
18 views

Is there an analogue of the 15 theorem for cubic forms?

The 15 theorem states that if an integral quadratic form with integral matrix represents the numbers 1, 2, 3, 5, 6, 7, 10, 14, 15, then it represents all numbers. Is there an analogue of this theorem ...