All Questions
Tagged with nt.number-theory diophantine-equations
787 questions
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Can someone explain to a "newbie" of number theory how Matijasevič demonstrated the impossibility of hilbert’s tenth problem?
As the title stated, I'm an amateur in the number theory that has just approached hilbert's tenth problem and the demonstration given by Matijasevic, but I couldn't find much on it, and what I could ...
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205
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On Sums of powers II
In my previous question, I asked about nontrivial sums of four powers $a^m+b^n+c^k=d^l$, and whether the nature of the solutions depend on whether $\frac{1}{m}+\frac{1}{n}+\frac{1}{k}+\frac{1}{l}$ is ...
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147
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Chinese remaindering to solve solvable diophantine equations
Given a diophantine equation $$f(x_1,\dots,x_z)=0$$ where $f(x_1,\dots,x_z)\in\mathbb Z[x_1,\dots,x_z]$ is of total degree $d$ and each variable degree $d_i$ where $i\in\{1,\dots,z\}$ there is no ...
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Small linear relations in unbalanced diophantine equations from primitive Pythagorean triples
$r$ is parameter. Pick coprime $m,n\in[r,2r]$ with $mn$ even. Consider the Linear Diophantine Equation $$a^4u+b^4v+c^2z=0$$ where $a=m^2-n^2$, $b=2mn$ and $c=m^2+n^2$.
Is it true that there are ...
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221
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Number of integer solutions to quadratic polynomial with integer coefficients
It is known from for example Representations of Integers as Sums of Squares by Grosswald, E. that
$$|\{(n_1,n_2,\ldots,n_k)\in\mathbb{Z}^k: \ n_1^2+n_2^2+\cdots+n_k^2=N\}|\leq C_\varepsilon N^{\frac{...
- 193
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264
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On variants of the abc conjecture in terms of Lehmer means
In this post we denote the Lehmer mean of a tuple $\text{x}$ of positive real numbers as $$L_p(\text{x})={\sum_{k=1}^nx_k^p\over\sum_{k=1}^nx_k^{p-1}},$$
see the reference Wikipedia Lehmer mean.
The ...
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63
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Quadratic factors of $l_1(x,y)^3+l_2(x,y)^3+l_3(x,y)^3-n$
Related to sum of three squares and this question.
Let $l_1,l_2,l_3 \in \mathbb{Z}[x,y]$ and $n \in \mathbb{Z}$.
Assume that $n$ is not a cube and not twice cube.
Let $f=l_1(x,y)^3+l_2(x,y)^3+l_3(x,...
- 25.4k
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64
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On the number of solutions of the equation involving Pochhammer symbols $(n)_a\cdot(n)_b=(n)_c$, for integers greater than or equal to $2$
As paticular case of the equation involving Pochhammer symbols $$(n)_a\cdot(m)_b=(k)_c,$$
where the variables are positive integers, I've consider the case $n=m=k$ of previous equation, that is
$$(n)...
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133
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What about an alternative formulation for different prime constellations in the spirit of Suzuki's theorem for twin primes?
It is known that the twin prime conjecture is a special case of the $k$-tuple conjecture. See if you want the article with title k-Tuple Conjecture from the encyclopedia Wolfram MathWorld.
On the ...
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80
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Find solutions and get a first statement for these diophantine equations inspired in certain figurate numbers
Yesterday I was thinking in speculative relationships between certain figurate numbers, please see if you need the tables, and references from the article of the
encyclopedia MathWorld Figurate Number....
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134
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Two equations and a question related to a well-known conjecture from number theory
On the Wikipedia page, the Beal´s conjecture is stated as:
If $A^x+B^y=C^z$, where $A,B,C,x,y,z$ are positive integers with $x,y,z>2$, then $A$,$B$, and $C$ have a common prime factor.
I think ...
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96
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Elementary constraints for the solutions of $z^{n-2}y(y+z)=x^n$?
Related to FLT and this question.
For natural $n > 4 $ define the curve $C_n : z^{n-2}y(y+z)=x^n$.
$C_n$ has the trivial points with $x=0$ for all $n$.
The answer in the linked question shows ...
- 25.4k
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72
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Superfluousness of ET-type $I$ for ES-equation (?)
You may consult the following paper by Christian Elsholtz & Terence Tao: https://terrytao.wordpress.com/tag/erdos-straus-conjecture/
A natural solution $\ (p\ x\ y\ z)\ $ of Erdös-Straus equation
...
- 7,500
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173
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The Diophantine equation $x^2 + bxy + cy^2 = p^z_1 \cdots p^{z_k}$
Let $b,c \in \mathbb{Z}$ and let $p_1,\ldots,p_k$ be given primes. Is there an effective algorithm to find all the solutions of the Diophantine equation $$x^2 + bxy + cy^2 = p_1^{z_1} \cdots p_k^{z_k}$...
- 11
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363
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On unique solutions to linear diophantine equations
Let $\sum_{i=1}^k a_ix_i = N$ be the equation with $a_i \in [2^t,2^{t+1}]$ being distinct primes. If we seek unique solutions $x_i\in R_i = (0,a_i)\cap \Bbb Z$, then in general it is not possible.
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- 13.9k
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292
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Exponential Diophantine $\prod x_i^{e_{x_i}} + \prod y_i^{e_{y_i}} = \prod z_i^{e_{z_i}}$ ,$e_{x_i}>3,e_{y_i}>3,e_{z_i}>3$
Does the exponential diophantine equation
$$ \prod x_i^{e_{x_i}} + \prod y_i^{e_{y_i}} = \prod z_i^{e_{z_i}}$$
with $x_i, y_i,z_i$ coprime and $e_{x_i}>3,e_{y_i}>3,e_{z_i}>3$ have solutions?...
- 25.4k
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322
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When is $Pn^2-2an+\frac{a^2-k}{P}$ , with $P$ Prime, $k=a^2 mod P$, a square?
It is easy to show that the following problems are equivalent.
a. When is $Pn^2-2an+\frac{a^2-k}{P}$ , with $P$ Prime, $k=a^2 mod P$, and $n$ any integer, a square?
and
b. When is $X^2-PY^2=k$ ...
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1
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126
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Proving that there are no integral points on a union of hyperbolas
I have a curve C: (x^2±x-y^2+1)(x^2∓x-y^2) where x,y ∈ Z+ that I want to prove has no non-trivial integral points other than (0,0),(1,0),(0,±1). I am having a hard time coming up with a solution.
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Write $n^2$ as $x^2+y^2+2\times4^z$ or $x^2+y^2+5\times 4^z$
In March 2018, I formulated the following somewhat curious question.
Question 1. Whether for any integer $n>1$ there is a nonnegative integer $k$ such that $n^2-2\times 4^k$ or $n^2-5\times 4^k$ ...
- 15.6k
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2
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642
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On the equation $x^3 + y^3 = z^4$
Are there any rational numbers $x, y, z$ with $xyz \neq 0$ and coprime numerators such that $x^3 +y^3 = z^4$ ?
user123305
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2
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270
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On the equation $a^{2}b^3 + c^{2}d^3 = e^{2}f^3$
Do there exist positive integers $a, b, c, d, e, f$ such that $a^{2}b^3 + c^{2}d^3 = e^{2}f^3$ where $b, d, f$ are pairwise coprime ?
Addendum: From the comments and Matt. F's answer, there clearly ...
- 1,019
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1
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365
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Positive integers written as $\frac{a(a+1)}2+\frac{b(b+1)}2+4^c5^d$
Let $\mathbb N=\{0,1,2,\ldots\}$. Those
$T_n:=n(n+1)/2$ with $n\in\mathbb N$ are called triangular numbers. It is well known that
$$\{T_a+T_b+T_c:\ a,b,c\in\mathbb N\}=\mathbb N\tag{1}$$
which was ...
- 15.6k
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1
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149
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Find the diophantine-equations $3x(x^2+2)=y^2$ integer solution [closed]
Let $x,y$ be positive integers, such that
$$3x(x^2+2)=y^2$$
since
$$3\cdot 1(1^2+2)=3\times 3=9=3^2$$
$$3\cdot 2(2^2+2)=6\cdot 6=36=6^2$$
$$24\cdot 3(24^2+2)=72\cdot 578=204^2$$
so I have ...
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1
answer
96
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Diophantine equation $546\cdot p+546\cdot q=1001\cdot r$ [closed]
$546\cdot p+546\cdot q=1001\cdot r$
$p,q$ odd primes, r positive integer.
are there infinitely many solutions?
And what if r is a Catalan number?
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2
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148
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Mordell like equation [closed]
This looks like a mordell like equation
X²=Y³-25056
How to solve it?
The exact equation is
(36x)²=(6y)³-25056
Is there any website has records of the equation x²=y³+k
For k>25000
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1
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Why wolfram alpha gives integers solutions for some equations of the form $ x^3 +(k\times10^n)^3 + z^3=0 $?
I have tried to get representations of some integers as sum of three cubic of the form $x^3+(k*10^n)^3+z^3$ with $k$ is integer and $n$ is a postive integer, I took this example : $(48807585839879)^3-(...
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1
answer
168
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Diophantine equation $10^n-a^3-b^3=c^2$
Consider the Diophantine equation:
$10^n-a^3-b^3=c^2$, for $a$, $b$, $c$, and $n$ positive.
Has this equation infinitely many solutions?
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1
answer
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When is the following fraction an integer $\frac{3^a}{2^b-3}[(\frac{2^b}{3})^c - 1]$ where $a,b,c \in \mathbb{Z}$? [closed]
I'm trying to evaluate the following fraction $\frac{3^a}{2^b-3}[(\frac{2^b}{3})^c - 1]$, but I'm getting stuck using gcd arguments or divisibility arguments. This is part of an ongoing research I'm ...
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4
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Finding integer zeroes for a particular family of equations [closed]
Given $p,q\in\mathbb Z^+$, and a vector $v=(x_1,\dots,x_{p+q})$ we consider the function $\chi(v)$:
$$\chi(v)=x_1^2+\dots+x_p^2-x_{p+1}^2-\dots-x_{p+q}^2$$
We wish to find solutions to $\chi(v)=0$ ...
- 1,226
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1
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201
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Solutions to a diophantine system
What is the smallest $\gamma_1,\gamma_2,\gamma_3>0$ such that given coprime $p,q=\Theta(\ell)$ and integer $t\geq3$ there are coprime $m,n=\Theta(\ell^{t-1})$ with $(mn,pq)=1$, $\alpha_i\in\Bbb Z$ ...
- 13.9k
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1
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$p=4x^2+27y^2$,with $p$ a prime [closed]
p is a prime ,on what condition the Diophantine equation is solvable.what is it Linear expression ,for example ,$x^2+3y^2=p$, $p=3k+1$ ,$x^2+5y^2=p$ ,
$p=1,9\pmod{20}$.
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2
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Non-vanishing of this ternary quadratic expression [closed]
I'm dealing with the expression $x^2+y^2+6z^2+8xy+4x+4y−6xz−6yz$. I want to show that this expression is always non-zero whenever $x,y$ and $z$ are positive integers. How does one do this? (Note that ...
- 1,109
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2
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608
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Rational points on the elliptic curve $y^2 = x^{3} - t^{2}z^3$
What are the rational points on the elliptic curve $y^2 = x^3 - t^{2}z^3$ ? I seem not to find any besides the trivial ones whereby $txyz=0$ or $x= \pm z$.
ADDENDUM 1. I have just noticed that if $z^3 ...
- 1,019
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2
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272
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Has nontrivial solution in positive integers of a diophantine equation: $x_1^2+x_2^2+x_3^2+x_4^2=y_1^2+y_2^2+y_3^2+y_4^2$ [closed]
Has nontrivial solution in positive integers of a diophantine equation as follows ?
$$x_1^2+x_2^2+x_3^2+x_4^2=y_1^2+y_2^2+y_3^2+y_4^2$$
Where trivial solutions are $x_i=y_j$.
Can you send me any ...
- 477
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1
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201
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Does the equation $x^k+y^k-z^k-w^k=3\ (k>3)$ have a solution over $\mathbb N$?
Clearly,
$$3=0^2+2^2-1^2-0^2\ \ \mbox{and}\ \ 3=4^3 +4^3-5^3-0^3.$$
Question. Let $k>3$ be an integer. Does the equation
$$ x^k+y^k-z^k-w^k=3\quad \ (x,y,z,w\in\mathbb N=\{0,1,2,\ldots\})\tag{1}$$
...
- 15.6k
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2
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233
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If $~(c - b) ^ 2 + 3cb = a^3~$ has nonzero integer solutions, then $~(a,c) \gt 1~$ or $~(b,c) \gt 1$? [closed]
If $~(c - b) ^ 2 + 3cb = a^3~$ has nonzero integer solutions, then $~(a,c) \gt 1~$ or $~(b,c) \gt 1$?
I think this is true, how to prove this?
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1
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150
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On Mordell equation $y^2=x^3+k$ [closed]
Have the Mordell equation $y^2=x^3+k$ solved for all integer $k$ or not?
Please Could you tell me about a good review papers about such equation.
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