Questions on finding integer/rational solutions of polynomial equations.

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48 views

Exponential diophantine: $2^x-7^y=z^2$

Find all integers $x,y,z$ such that $2^x-7^y=z^2$. For example: $2^3-7^1=1^2$ $2^5-7^1=5^2$ $2^7-7^1=11^2$ (But note that $\sqrt{2^9-7^1}\not\in \mathbb{Z}$.) The problem with this particular ...
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50 views

Large initial solutions to $x^3+y^3 = Nz^3$?

Let $x,y,z$ be non-zero integers. Is it true that the initial or smallest solution (in terms of absolute value) to, $$x^3+y^3 = Nz^3\tag1$$ for $N=94$ is, $$15642626656646177^3 + ...
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37 views

About Runge's method

I have been reading about some Diophantine equations (like Runge's theorem and Cassel's theorem) and in the text says that these theorems are solved using Runge's method, but it doesn't say what ...
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66 views

Transforming the cubic Pell-type equation for the tribonacci numbers

The Lucas and Fibonacci numbers solve the Pell equation, $$L_n^2-5F_n^2=4(-1)^n\tag1$$ The tribonacci numbers $z = T_n$ are positive integer solutions to the cubic Pell-type equation, $$27 x^3 - 36 ...
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71 views

The number of solution of a Diophantine equation

If we fixe $n\in \mathbb{N}$. I was wondring if there is an estimation of the number of the integer solutions of the equation : $$x_1^2+x_2^2+\cdots+x_n^2=n^3 $$ where $x_i>0$ for all ...
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72 views

How to solve this equation $x^5 +4^y=2013^z$ in positive integers?

I think to solve the equation in positive integers. It appears in a contest and I don't remember where. I obtain that $x$ must be an odd number and further $x=1 \, mod\, 4$. Any hint is appreciated.
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47 views

Properties of non-equivalent solutions to the generalized Pell equation

Given the Diophantine equation $$ r^2-ds^2 = x^2-dy^2 = q, $$ (where $q$ is a potentially unknown integer, and certainly need not be $1$), the two solutions $(r,s)$ and $(x,y)$ are called equivalent ...
2
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69 views

Diophantine $7^a+2=3^b$

I want to find the solutions $(a,b)\in\mathbb{Z}^+\times\mathbb{Z}^+$ of $7^a+2=3^b$. One such solution is $(a,b)=(1,2)$. Looking modulo $4$, we have $(-1)^a+2\equiv(-1)^b$, so $a$ and $b$ are of ...
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51 views

Quadruples of integers with $20^x + 14^{2y} = (x + 2y + z)^{zt}.$

Determine all quadruples $(x,y,z,t)$ of positive integers such that $$20^x + 14^{2y} = (x + 2y + z)^{zt}.$$ We can check that $20+14^2=216=(1+2+3)^3$. But how can we check if there are other ones?
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92 views

Is Legendre’s solution of the general quadratic equation the only one?

Legendre famously solved the general quadratic equation $$ ax^2+bxy+cy^2+dx+ey+f=0 $$ by noting that \begin{equation*} 4a(b^2-4ac)(ax^2+bxy+cy^2+dx+ey+f) = 0 \tag{$\star$} \end{equation*} along with ...
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99 views

How to solve $x^2+11=y^3$?

I've been trying to solve the diophantine $$x^2+11=y^3$$ recently but to no avail. I tried the "UFD trick", re-writing as $(x-i\sqrt{11})(x+i\sqrt{11})=y^3$, but it didn't give me all the solutions. I ...
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55 views

The Diophantine equation $x^n - y^n = z^2$

Darmon-Merel theorem (DMT) ensures that if $n \geq 4$ is an integer and $x, y, z > 0$ are integers such that $(x, y, z) = 1$ then $x^n + y^n \neq z^2.$ The question is: Does DMT apply to the ...
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73 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, ...
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67 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 ...
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87 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 ...
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76 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 ...
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83 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 ?
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49 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 ...
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118 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 ...
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102 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 ...
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110 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 ...
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149 views

The equation $y^2-x^t=k$.

S.E board! This is a Diophantine equations problem, which is so interesting one can do by plugging the suitable values in unknown. When it comes for finding set of all solutions is may be tough. I ...
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109 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 ...
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205 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 ...
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33 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 ...
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415 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: ...
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143 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 ...
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31 views

The exponent on Thue's theorem

I have been reading about Runge's theorem on diophantine approximation Theorem. Let $\xi$ be an algebraic real number of degree $d\geq 3$. For every $\epsilon >0$ there is a number $\gamma >0$ ...
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30 views

Diophantine linear Equation Gaussian Integers

We know that $ax+by=c$ with $gcd(a,b)=1$ could be solved over $\Bbb Z$. Supposing if $a,b,c\in\Bbb Z[i]$, is there an analogous framework to find $x,y\in\Bbb Z[i]$ (at least of minimum norms)?
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36 views

How to solve $xy+ax+by+c=0$ in inetegrs?

Respected all. Before I ask your support, let me show you what I have done and have got stuck. We are willing to solve $2x+3xy+4y=5$. So this is what I have done. The given equation becomes ...
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34 views

Other Diophantine problems that use a Pell equation

What Diophantine equations employ Pell equations in their solutions? A well-known example is the case of Pythagorean triples where the legs differ by 1, like, $$20^2+21^2 = 29^2$$ These are ...
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32 views

Hensel's lemma in $n $ variables

I'm trying to find a proof for the following formulation of Hensel's lemma: $$\text{Let } f \in \mathbb{Z}[x_1, \dots, x_n], a = (a_1, \dots, a_n) \text{ be such that (with } p \text{ prime)}$$ $$ ...
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71 views

Form of solutions Pell's equation

I'm studying a proof regarding Pell's equation. It has the form $y^2 - Dx^2 = 1$ with $D \in \mathbb{N}$. Namely that it has an infinite number of solutions if $D$ is not a perfect square. I already ...
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53 views

An elliptic curve for the multigrade $\sum^8 a_n^k = \sum^8 b_n^k$ for $k=1,2,3,4,5,9$?

I. The first solution to, $$\sum^6_{n=1} a_n^9 =\sum^6_{n=1} b_n^9$$ $$13^9+18^9+23^9-5^9-10^9-15^9 = 9^9+21^9+22^9-1^9-13^9-14^9$$ was found in 1967 by computer search by Lander et al. It stood ...
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48 views

System of linear diophantine modular inequalities

How can we best find a numerical solution to a system of $m\ge2$ linear diophantine modular inequalities $$\big((a^j x+b_j)\bmod n\big)<c\;\text{ for }1\le j\le m$$ where $x$ is the only unknown, ...
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47 views

Solve the Diophantine equation $y^3=4x^2+4x+5$ in $x,y\in\mathbb{Z}$

I have to solve the Diophantine equation $y^3=4x^2+4x+5$ where $x,y\in\mathbb{Z}$ and I have been thinking now for a long time and I have really no clue how to do this. The only hint given in the ...
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42 views

Theorem for Equal Sums of Like Powers $x_1^8+x_2^8+x_3^8+\dots$

Kindly see the question at the end of post. Solutions to the system of three equations, $$\begin{aligned} a^2+b^2+c^2+d^2\, &= e^2+f^2+g^2+h^2\\ a^4+b^4+c^4+d^4\, &= e^4+f^4+g^4+h^4\\ abcd\, ...
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73 views

Does this system of simultaneous Pell equations have any non-trivial positive integer solutions?

Let $a,b,c$ be positive integers satisfying \begin{align} 2a^2-1 &= b^2, \\ 2a^2+1 &= 3c^2. \end{align} The trivial solution is $(a,b,c)=(1,1,1)$. Are there others?
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43 views

Generalization of a Diophantine Equation Problem

I've been working a lot with Pythagorean triples and sums of squares that are themselves squares, specifically interlocking sums (where one square is part of two or more sums). As part of my work I ...
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53 views

The Diophantine equation $ax+by = p^z$.

Let $a,b \in \mathbb{Z}$ and let $p$ be a prime. Let $d = \gcd(a,b)$, and let $s$ and $t$ be integers such that $d = as+bt$. The solutions $(x,y,z) \in \mathbb{Z} \times \mathbb{Z} \times ...
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29 views

Why do we conclude that $(a,b)=1$, having found that $(a',b'=1)$?

Suppose that we have the equation $ax^2+by^2+cz^2=0, a,b,c \in \mathbb{Q}$. Without loss of generality, we suppose that $gcd(a,b,c)=1$. Also, we can consider that $a,b,c$ are square-free. We can ...
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64 views

When $Ax^2+By^2=z^2$ has a solution in integers?

Consider the Diophantine equation $Ax^2+By^2=z^2$, with positive integer parameters $A$ and $B$ (not necessarily square-free or co-prime). When does this equation have a non-trivial solution? Can one ...
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42 views

Siegel's theorem and singular curves

I notice that often Siegel's theorem (there are only finitely many integral points on a curve of genus greater than 0) is stated with the requirement that the curve be smooth. Other times the ...
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74 views

Woking Heron's Formula In Reverse

I'm writing a program to generate randomized Heron's Formula word problems. I need to figure out how to work the problem in reverse so that the answer will come out to an integer. As an example, if I ...
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123 views

Any general “formula” solutions for higher order polynomial equation?

We know that fifth (or higher) degree polynomial equation has no general solution formula using only the usual algebraic operations (addition, subtraction, multiplication, division) and application of ...
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60 views

Integral point on certain cubic surfaces and rational parametric solutions

The motivation of the following question comes from the Problem D4 in the book "unsolved problems in number theory" by Richard Guy. No integral points on the surface $S:x^3+y^3+z^3=3$ is known other ...
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42 views

Powerful numbers in Pell solutions (or, more generally, any Lucas sequence)

There are several definitive results regarding perfect powers in the Pell numbers — e.g., the only perfect power is $P_7=169=13^2$. On the other hand, when it comes to powerful numbers, I've only ...
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44 views

Given a Pell “solution” in [integer] polynomials, what can be said about the components?

Let $x,y$ be integers, and $f(x,y)$, $g(x,y)$, and $h(x,y)$ be polynomials in $x$ and $y$ with integer coefficients such that $$ f(x,y)^2 - g(x,y)h(x,y)^2 = 1. \qquad(\star) $$ Furthermore, assume it ...
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29 views

Characterizing Coprimes

Here's a question about coprimes that I stumbled upon while doing some research. Providing insight into this question would prove quite helpful to me. Choose a pair of coprimes $x, y \in \mathbb Z$. ...
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42 views

Differential Diophantine Equations?

So this is both a question on its own as well as a request for where I can find information on a given topic. Consider Differential Equations in two variables of the form: $$P(Z,Z', Z'' ... Z^{[n]}, ...