Questions on finding integer/rational solutions of equations.
0
votes
0answers
8 views
Combinatorics of the Zeta function of a variety
I want to know if there is a good combinatorial interpretation of what the Zeta function of a variety $X$ over a finite field $\mathbb{F}_p$ counts. It is defined as $$\exp\sum N_j/j\,t^j,$$ where ...
2
votes
1answer
64 views
Sums of powers being powers of the sum
I'm looking for literature on solving problems of the form
$$
n_1^\alpha+\cdots+n_k^\alpha=(n_1+\cdots+n_k)^\beta
$$
for positive integers $n_1,\ldots,n_k$ and fixed parameters $k$ and ...
16
votes
7answers
1k views
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 ...
0
votes
2answers
25 views
Solving a system of Diophantine equations
For a problem that I'm working on, I need to solve the following system of Diophantine equations:-
$a^3+40033=d$, $b^3+39312=d$, $ c^3+4104 = d$ where $a,b,c>0$ are all DISTINCT positive integers, ...
2
votes
4answers
433 views
solve $100x - 23y = -19$
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 to ...
6
votes
1answer
98 views
Help solving this Diophantine equation
For a problem that I'm working on, I need to solve this Diophantine equation:-
$ -2a^3 + b^3 + c^3 = 36650$, where $a, b, c > 0$ are all DISTINCT positive integers, and $a, b, c \notin$ { 2, 9, ...
3
votes
1answer
43 views
System of Diophatine equations $x_1,\ x_2,\ x_3,\ \ldots,\ x_n,\ y \in \mathbb{Z^+}$
Let $a_1,\ a_2,\ a_3,\ \ldots,\ a_n$ be distinct positive integers.
Find $x_1,\ x_2,\ x_3,\ \ldots,\ x_n,\ y \in \mathbb{Z^+}$
such that: $$\left\{\begin{array}{rl}(x_1,x_2,\ldots,x_n)&=1\\ ...
6
votes
3answers
135 views
How many solutions are possible to this equation?
Given
$$A+2B+3C=N
$$
where $N$ is a given positive integer.
$A ,B,C\in\mathbb{N}$ vary from $0$ to $\infty$.
How many solutions will be there to this equation?
3
votes
1answer
56 views
How to solve $a_{1} + 2a_{2} + 3a_{3} +\cdots + (n-1)a_{n-1} = n $
How do I generally solve
$a_{1} + 2a_{2} + 3a_{3} +\cdots + (n-1)a_{n-1} = n $ where $a_{i} $ are nonnegative integers?
2
votes
2answers
54 views
On the Pell-like $px^2-qy^2 = 1$ for prime $p,q$
Given any prime of form $p_n = u^2+nv^2$ for non-zero integers $u,v$. Consider,
\begin{aligned}
&p_2x^2-2y^2 = 1\\
&p_3x^2-3y^2 = 1\\
&p_7x^2-7y^2 = 1\\
&p_{11}x^2-11y^2 = 1\\
...
13
votes
2answers
230 views
How did Letac solve $x_1^k + x_2^k + \dots +x_9^k = 0$ for $k = 1, 3, 5, 7$ in 1942?
It's quite easy to find integer solutions to,
$$x_0^k + x_1^k + \dots +x_9^k = 0$$
for $k = 1, 3, 5, 7$. One I found is, if $x^2-10y^2 = 9$, then,
$$1 + 5^k + (3+2y)^k + (3-2y)^k + (-3+3y)^k + ...
0
votes
3answers
26 views
General Solution of Diophantine equation
Having the equation:
$$35x+91y = 21$$
I need to find its general solution.
I know gcf $(35,91) = 7$, so I can solve $35x+917 = 7$ to find $x = -5, y = 2$. Hence a solution to $35x+91y = 21$ is ...
3
votes
0answers
53 views
On Sixth Powers $x_1^6+x_2^6+\dots+x_6^6 = z^6$
Fourteen years ago, in 1999 (has it been that long?) Merignac started a search for,
$$x_1^6+x_2^6+\dots+x_6^6 = z^6$$
using the five congruence classes,
$$\begin{aligned}
...
5
votes
2answers
54 views
About the infinite solutions of a Diophantine equation
Consider the following problem:
$$\sum_{k=1}^N k^2=q^2$$
where q is an integer number. This can be written as:
$$\frac{1}{3}N^3+\frac{1}{2}N^2+\frac{1}{6}N=q^2$$
In the same way we can write:
...
0
votes
1answer
43 views
Solve a linear equation with 3 unknowns and 1 parameter
$$(a+1)x+y+3z=1$$$$8x+2y+(a+3)z=2$$$$3x+y+2z=-1$$
This question can be calculated with Gauss-elimination and I want to take away the y by taking $-2$ from the middle and $-1$ from the top and keep ...
1
vote
1answer
41 views
A Diophantine equation and decimal digits
Solutions of the Diophantine equation
$a10^n+(a+1) = (2^{m+1}-1)*2^{m+1}$
are
12=3*4,
56=7*8,
67100672=8191*8192.
Are there more solutions/examples like that or a generalization of the ...
2
votes
1answer
63 views
Total no. of ordered pairs $(x,y)$ in $x^2-y^2=2013$
Total no. of ordered pairs $(x,y)$ which satisfy $x^2-y^2=2013$
My try:: $(x-y).(x+y) = 3 \times 11 \times 61$
If we Calculate for positive integers Then $(x-y).(x+y)=1.2013 = 3 .671=11.183=61.33$
...
-1
votes
2answers
57 views
Context problems of Number theory and functional equation
I can't solve the following problems, please help.
1) Find all primes $p$ and $q$ such that $p^q+q^p$ is a prime.
2) Solve $2^x+3^y=z^2$ in integers.
3) Find all $f: \mathbb{Q} \rightarrow ...
9
votes
1answer
75 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 ...
2
votes
1answer
35 views
Solution of a Diophantine equation involving powers
Is it possible to show that, given $n$, there are infinite values of $k$ giving solutions of the equation:
$$x^n+ky^n=z^n$$
with $k,x,y,z,n$ natural numbers?
The constrains are: $$2\lt n, 1\lt k$$
6
votes
2answers
137 views
Solve: $x^2-py^2=q$
Solve $$x^2-py^2=q$$ for integers $x,y$, here $p,q$ are both given prime numbers.
It's obvious that $p,q$ should satisfy $(\frac{p}{q})=(\frac{q}{p})=1,$ here $(\frac{p}{q})$ is the Jacobi symbol.
...
7
votes
2answers
169 views
Solve : $\space 3^x + 5^y = 7^z + 11^w$
Solve the diophantine equation $3^x + 5^y = 7^z + 11^w$,here $x,y,z,w$ are all non-negative integers.
I find three solutions by force algorithm use Mathematica: (0,0,0,0)(1,1,1,0)(1,3,1,2).And ...
7
votes
2answers
105 views
For which integers x, y is $2^x + 3^y$ a square of a rational number?
For which integers x, y is $2^x + 3^y$ a square of a rational number?
(Of course $(x,y)=(0,1),(3,0)$ work)
2
votes
2answers
114 views
Find all answers of $n^2-2^m=1$
Find all natural numbers $(n,m)$ where $n^{2}-2^{m}=1$.
I have my own answer of that, however I wanted to know if anyone has a better or easier answer or not!
3
votes
0answers
43 views
Generalizing Ramanujan's 6-10-8 Identity
Ramanujan's 6-10-8 Identity can be succinctly given. Define,
$$F_k = a^k+b^k+c^k-(d^k+e^k+f^k)$$
If $F_2 = F_4 = 0$ and $a+b+c = d+e+f = 0$, then Ramanujan found that,
$$64F_6 F_{10} = 45F_8^2$$
...
11
votes
2answers
206 views
How to find a “better description” (e.g. recurrence relation) for this sequence?
My solution to a problem in Project Euler required to solve this subproblem: find values of $k\in\mathrm{N}$ such that $3k^2+4$ is a perfect square.
As I was writting a computer program, I just tried ...
3
votes
1answer
113 views
Find all integer solutions to $x^2+4=y^3$.
Find all integer solutions to $x^2+4=y^3$.
Some obvious solutions are $(x,y)=(\pm2,2)$. Are these the only ones?
4
votes
2answers
61 views
Foundation on Diophantine Analysis and Number Theory
I want to read particularly about diophantine Analysis and Elementary Number Theory from a novice level.
The books which I found on net:
A Guide to Elementary Number Theory by Underwood Dudley
...
1
vote
3answers
65 views
Way to show $x^n + y^n = z^n$ factorises as $(x + y)(x + \zeta y) \cdots (x + \zeta^{n-1}y) = z^n$
For odd $n$ the Fermat equation $x^n + y^n = z^n$ factorises as $$(x + y)(x + \zeta y) \cdots (x + \zeta^{n-1}y) = z^n,$$
where $\zeta = e^{2 \pi i/n}$. I tried seeing this was true by multiplying ...
151
votes
13answers
17k views
Unusual 5th grade problem, how to solve it
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
1answer
82 views
(USAJMO)Find the integer solutions:$ab^5+3=x^3,a^5b+3=y^3$
Find the integer solutions:
$$a·b^5+3=x^3,a^5·b+3=y^3$$
This is the first problem of today's USAJMO (has finished),I only find a trival result that $x\equiv y \pmod6$ and $abxy≠0 \pmod 3$.
Thanks in ...
0
votes
2answers
144 views
Quadratic equations
Does anyone know how to find integer solutions of the quadratic equation
$$y^2+y+z=f$$
where $z$ is a fixed odd prime or $1$ and $f$ is a fixed odd prime greater than $3$?
This problem arose from ...
27
votes
6answers
570 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
$$
\begin{equation}
\left( 26+15\sqrt{3}\right) ^{1/3}+\left( 26-15\sqrt{3}\right) ^{1/3}=4.\tag{1}
\end{equation}
$$
...
0
votes
0answers
92 views
The equation $( m^2 + n^2 + q^2 )^2 = 36 ( u^2 + s^2 + t^2 )$
What is known about solutions in integers of the following equation ?
$$( m^2 + n^2 + q^2 )^2 = 36 ( u^2 + s^2 + t^2 )$$
I am asking this because I just recently have got these:
$$(a-b)^2 + ...
9
votes
1answer
153 views
How to find all integers $a,b > 1$ satisfying $b \mid a^2+1$ and $a^2 \mid b^3+1$?
Let $a,b\in \mathbb{Z}$ with $a,b>1$, and such that $b \mid a^2+1$ and $a^2 \mid b^3+1$. Find all such $a,b$.
I found $a=3,b=2$. Are there any other solutions? Thank you.
yesterday I have ...
0
votes
1answer
43 views
find all positive integers for a given diophantine equation involving 4 or 7 variables
Given equation:
Ap + Bq + Cr + Ds + Et + Gu + Vg = K; (Eq. in 7 variables);
suppose we have A, B, C, D initialize with = 1,2,5,10,20,50 and 100 respectively; and K = 50000;
How do we solve it?
...
2
votes
2answers
268 views
Finding a basis for the solution space of a system of Diophantine equations
Let $m$, $n$, and $q$ be positive integers, with $m \ge n$.
Let $\mathbf{A} \in \mathbb{Z}^{n \times m}_q$ be a matrix.
Consider the following set:
$S = \big\{ \mathbf{y} \in \mathbb{Z}^m \mid ...
0
votes
1answer
43 views
Linear Diophantine Equations: Integer Solutions $x,y$ exist for $ax+by=c$, but how do I find them by hand?
I'm trying to find which of $133x+203y=38$, $133x+203y=40$, $133x+203y=42$, and $133x+203y=44$ have integer solutions. I know that only the third equation suffices for these conditions because ...
3
votes
1answer
460 views
Integer coordinate set of points that is a member of sphere surface
I have a graphic application to develop which involve many spheres. I should determine then on run time.
Supposing that I have a sphere of radius r, how can I determine the sub set of the sphere ...
4
votes
3answers
77 views
How to solve for solutions to this diophantine?
I have the diophantine equation $y(x+y+z) = xz$ where all variables are positive integers. Given some bound $y \leq B$, how can I count the number of solutions?
3
votes
2answers
68 views
Factorials and Arithmetic Progression.
Are there sets of factorials $(a_1!,a_2!,a_3!,\dots,a_n!)$, such that they exist in Arithmetic progression.
$n$ is a natural number
I don't see any such examples(Except for $n=2$). And I don't see ...
2
votes
2answers
46 views
Sums that are pythagorean and normal
I noticed that
$3^2+4^2+15^2=9^2+13^2$
and also
$3+4+15=9+13$
Is there an easy way to find all pairs of disjoint sets of positive integers whose sum are the same and whose sum of squares are the ...
2
votes
1answer
50 views
Simple looking diophantine equation…
I recently found myself asking if the following (diophantine) expression ever evaluates to a square number:
$$5+12n$$
I was surprised both to be unable to stumble across an integer value for $n$ ...
4
votes
1answer
46 views
On Selmer's curve
I am trying to prove that the equation $3x^3+4y^3+5z^3 \equiv 0 \pmod{p}$ has non-trivial solutions for all primes $p$. I divide it into 3 cases: $p \equiv 0,1,2 \pmod{3}$. The cases $p \equiv 0,2 ...
5
votes
3answers
155 views
Integer solutions of $n^3 = p^2 - p - 1$
Find all integer solutions of the equation, $n^3 = p^2 - p - 1$, where p is prime.
0
votes
1answer
29 views
3 equations with 9 unknown variables with scalar product
Excuse my bad english pls. I can't find a proper solution to my problem because i don't know the exact mathematical terms in english.
My problem is how to get the 3 elements of each of 3 vectors ...
0
votes
2answers
43 views
Equation with matrix
$$
\begin{pmatrix}
3 & -1\\
-4 & 2 \\
\end{pmatrix}=(X^T+3I)^{-1}
$$
$T$ is the transpose and $-1$ is the inverse and $I$ is the unity matrix.
I have come this ...
2
votes
2answers
171 views
rational triangles and cosines
I've recently started to try working on exercises from this book on Diophantine equations before I need to return it to the library. This one has me slightly stumped. It asks to show that the cosine ...
0
votes
0answers
43 views
Wrong answer on elementary diophantine equation - why?
Solve the equation and show all possible, non-negative values for X
and Y: $5X+4Y=60$
So I wanted to do it like that:
$$5X+4Y=60\leftrightarrow0X+4Y=0 \pmod5$$, thus $4Y=5k$ where $k\in Z$. ...
2
votes
1answer
43 views
Number of integer solutions of $xy - 6 (x+y)=0$
What are the number of integer solutions of $xy - 6 (x+y)=0$ with $x\leq y$ is ?
Equation $xy - 6 (x+y)=0$ can also be written as $1/x + 1/y = 1/6$
