All Questions
Tagged with nt.number-theory diophantine-equations
787 questions
3
votes
1
answer
217
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Does the equation in positive integers $(n,y)\,\prod_{k=1}^n(p_k^2-1)=y^2\,$ only have the solution $(3,24)$?
Does the equation in positive integers $\,(n,\,y)$
$$\prod_{k=1}^n(p_k^2-1)=y^2$$
only have the solution $(3,\,24)\,$?
I asked a more general question here.
The computational complexity of the problem ...
- 1,008
10
votes
0
answers
205
views
Does the diophantine equation $\,\prod_{k=1}^n(p_k^{x_k}-1)=y^2\,$ have always at least a solution for $\,n\gt2\,$?
P.G.Walsh proved in this paper that the diophantine equation $\,(2^{x_1}-1)(3^{x_2}-1)=y^2\,$ has no solution in positive integers $\,x_1$, $\,x_2\,$ and $\,y$.
If we generalize the previous equation ...
- 1,008
9
votes
2
answers
644
views
Diophantine equation $3(a^4+a^2b^2+b^4)+(c^4+c^2d^2+d^4)=3(a^2+b^2)(c^2+d^2)$
I am looking for positive integer solutions to the Diophantine equation $3(a^4+a^2b^2+b^4)+(c^4+c^2d^2+d^4)=3(a^2+b^2)(c^2+d^2)$ for distinct values of $(a,b,c,d)$.
There are many solutions with $a=b$ ...
- 506
8
votes
0
answers
271
views
Restricted divisor summatory function
I have a problem that boils down to prove that the succession $\{a_n\}$ tends to infinity, where
$$a_n:=1+\sum_{0\leq j<n}D_{2j+1}(n-j)$$
and $D_{m}(n)$ is the number of divisors $d>1$ of $n$ ...
- 361
5
votes
0
answers
256
views
Are the numbers $\varphi(n^2)\sigma(n^2)\ (n=1,2,3,\ldots)$ pairwise distinct?
Let $\varphi$ be Euler's totient function, and let $\sigma(n)=\sum_{d\mid n}d$ for $n=1,2,3,\ldots$. Both $\varphi$ and $\sigma$ are multiplicative functions. It is easy to see that the numbers
$$\...
- 15.6k
4
votes
1
answer
675
views
How to solve this equation $a^2+3b^2c^2=7^c$
Let $a,b,c$ be poistive integers,and such $\gcd(a,b)=\gcd(b,c)=\gcd(a,c)=1$,fine the all $a,b,c$ such
$$a^2+3b^2c^2=7^c$$
I'm not sure that this question has been studied, but I've been trying for a ...
- 4,280
19
votes
1
answer
679
views
Is it true that $\{x^4+y^2+z^2:\ x,y,z\in\mathbb Z[i]\}=\{a+2bi:\ a,b\in\mathbb Z\}$?
Recall that the ring of Gaussian integers is
$$\mathbb Z[i]=\{a+bi:\ a,b\in\mathbb Z\}.$$
Clearly
$$(a+bi)^2=a^2-b^2+2abi\ \ \mbox{and}\ \ (a+bi)^4=(a^2-b^2)^2-4a^2b^2+4ab(a^2-b^2)i.$$
Question. Is it ...
- 15.6k
0
votes
0
answers
122
views
Genus $0$ algebraic curves integral points decidable?
It is known there is an explicit algebraic variety in $\mathbb Z[x_1,\dots,x_t]$ a bounded $t>2$ whose integral zero-set is non-empty is undecidable.
If the variety has genus $0$ is there anything ...
- 13.9k
0
votes
0
answers
183
views
A certain Pell Equation
Recently I came up with a positive solution $((x,y)\neq (\pm 1;0))$ to this diophantine equation
$$
x^2-\left(w^2(2^{n-2}p)^2+2^n(2^{n-2}p)\right)y^2=1,\qquad n\geq 2,
$$
where all variables are in $ ...
- 785
0
votes
0
answers
115
views
Maximum number of integer solutions with some size constraints to bivariate polynomials?
Take a bivariate polynomial of total degree $d$ satisfying $d=d_x=d_y>1$ in $\mathbb Z[x,y]$ with coefficients bound in absolute value by $b$ ($d_x$ is $x$-degree and $d_y$ is $y$-degree).
Given a ...
- 13.9k
1
vote
0
answers
88
views
Distribution of number of integer solutions in box to bivariate polynomials?
Take a bivariate polynomial of degree $d_x+d_y>\max(d_x,d_y)>1$ in $\mathbb Z[x,y]$ with coefficients bound in absolute value by $b$ ($d_x$ is $x$-degree and $d_y$ is $y$-degree).
What is the ...
- 13.9k
2
votes
3
answers
416
views
Density of $d$ for which a generalized Pell equation has a solution
For how many $0 < d \leq D$ is there an integer solution to
$$x^2-dy^2 = -n$$
for $n > 1$? I have circumstantial reason to believe it might be $\sim D^{\frac{1}{2}}$ but I'd be interested in any ...
- 479
1
vote
0
answers
192
views
Integer solutions of Diophantine equation $y^2= 1+4n^{\underline k} $
I am looking for the integer solutions for the diophantine equation $y^2 =4n(n-1)(n-2)\cdots (n-k+1)+1$ for a given $k$ where $n>k+1>5$.
In other words,
$$y^2=1+4n^{\underline k},\tag{I}$$
where ...
4
votes
0
answers
504
views
Collatz conjecture and a diophantine equation
Let $M \ge 2$. Inspired by the Collatz iteration / algorithm ($M=2$), I tried the following function:
$$C_M(n) = n/M, \text{ if } n \equiv 0 \mod M, \text{ otherwise } (M+1)n+\{(M-n) \mod M \}$$
We ...
- 3,064
6
votes
4
answers
487
views
When is $\{b^2 - \{b-1\}_2\}_2=1$ with odd $b$? (The bracket-notation explained below)
For the complete extraction of the factor $p$ and its powers from a natural number $n$
let's define the notation $$ \{n\}_p := { n \over p^{\nu_p(n)}} \tag 1$$
$ \qquad \qquad $ Here $\nu_p(n)$ means ...
- 5,342
2
votes
1
answer
164
views
a b c triples with bounded prime factors
(i) For any fixed $B>0$, are there only finitely many triples $a,b,c$ of coprime positive integers, such that $a+b=c$ and all prime factors of $a,b,c$ are at most $B$?
(ii) For which $B$ all such ...
- 743
3
votes
1
answer
547
views
$x^2+7y^2=2^n$ and sums of four squares
Lagrange's four square theorem states that each $m\in\mathbb N=\{0,1,2,\ldots\}$ can be written as a sum of four squares.
Recently, I found that the diophantine equation $x^2+7y^2=2^n$ has certain ...
- 15.6k
1
vote
1
answer
198
views
Analytically controlling sizes in modular arithmetic to demonstrate Dirichlet pigeonhole application
Given $a,b,c,d\in\mathbb Z$ with $ad+bc=p$ a prime there is an $m\in\mathbb Z$ with $-\lceil1+\sqrt{p}\rceil<
r_1,r_2<\lceil1+\sqrt{p}\rceil$ and
$$r_1\equiv mac\bmod p$$
$$r_2\equiv mbd\bmod p$$...
- 13.9k
0
votes
0
answers
110
views
Common integer roots of polynomials
I have two polynomials of form
$$f_1(w,x)=M_1$$
$$f_2(y,z)=M_2$$
and I have two polynomials of form
$$g_1(w,x,y,z)=M_3$$
$$g_2(w,x,y,z)=M_4$$
where $f_1,f_2,g_1,g_2\in\mathbb Z[w,x,y,z]$ and $M_1,M_2,...
- 13.9k
0
votes
0
answers
147
views
Can someone explain to a "newbie" of number theory how Matijasevič demonstrated the impossibility of hilbert’s tenth problem?
As the title stated, I'm an amateur in the number theory that has just approached hilbert's tenth problem and the demonstration given by Matijasevic, but I couldn't find much on it, and what I could ...
-1
votes
2
answers
642
views
On the equation $x^3 + y^3 = z^4$
Are there any rational numbers $x, y, z$ with $xyz \neq 0$ and coprime numerators such that $x^3 +y^3 = z^4$ ?
user123305
0
votes
1
answer
325
views
On the elliptic curve $y^2 = x^3 + z^{4k}$
Are there any rational numbers $x, y, z$ with $xyz \neq 0$ such that $y^2 = x^3 + z^{4k}$ for some $k \in \mathbb{Z}_{>1}$ ?
- 1,019
6
votes
0
answers
506
views
Does the equation $\sigma(\sigma(x^2))=2x\sigma(x)$ have any odd solutions?
This question was posted in MSE in early August 2020. It did garner several upvotes, but did not receive any responses. I have therefore cross-posted it here, hoping that it gets answered.
Let $\...
2
votes
1
answer
354
views
The Fermat-Catalan conjecture with signature $(2,n,4)$, $n\ge4$
The Fermat-Catalan conjecture is that for coprime $x,y,z$ and positive integers $a,b,c$ with $1/a+1/b+1/c<1$, the generalized Fermat equation $x^a + y^b = z^c$ has only finitely many solutions. I'm ...
- 141
-1
votes
2
answers
270
views
On the equation $a^{2}b^3 + c^{2}d^3 = e^{2}f^3$
Do there exist positive integers $a, b, c, d, e, f$ such that $a^{2}b^3 + c^{2}d^3 = e^{2}f^3$ where $b, d, f$ are pairwise coprime ?
Addendum: From the comments and Matt. F's answer, there clearly ...
- 1,019
4
votes
1
answer
909
views
How to prove that a diophantine equation has only finitely many solutions in integers?
In 1976 Tijdeman proved that the Catalan equation
$$
x^{p}-y^{q}=1
$$
has finitely many solutions in integers $x,y,p,q>1$ in his paper
R. Tijdeman, On the equation of Catalan, Acta Arith. 29 (1976)...
- 43
13
votes
1
answer
499
views
On the equation $a^6+b^6+c^6=d^2$
I have been studying the equation $a^6+b^6+c^6=d^2$, trying to find rational solutions. I know it is a K3 surface, with high Picard rank, so there should be rational or elliptic curves on it.
When ...
- 2,811
10
votes
5
answers
771
views
Reference request: Diophantine equations
I am looking for a textbook, or preferably lectures, on the subject of Diophantine equations. I am familiar with the basic principles of modular arithmetic, conics and the Hasse Principle, and the ...
- 2,811
0
votes
0
answers
205
views
On Sums of powers II
In my previous question, I asked about nontrivial sums of four powers $a^m+b^n+c^k=d^l$, and whether the nature of the solutions depend on whether $\frac{1}{m}+\frac{1}{n}+\frac{1}{k}+\frac{1}{l}$ is ...
- 2,811
4
votes
1
answer
501
views
On sums of powers
I was considering the Fermat Catalan conjecture, where the equation $a^m+b^n=c^k$ has only finitely many nontrivial solutions (with coprime $a, b, c$) with $\frac{1}{m}+\frac{1}{n}+\frac{1}{k}=1$ (and ...
- 2,811
-3
votes
2
answers
608
views
Rational points on the elliptic curve $y^2 = x^{3} - t^{2}z^3$
What are the rational points on the elliptic curve $y^2 = x^3 - t^{2}z^3$ ? I seem not to find any besides the trivial ones whereby $txyz=0$ or $x= \pm z$.
ADDENDUM 1. I have just noticed that if $z^3 ...
- 1,019
3
votes
1
answer
370
views
Close integer solutions to $ab-cd=1$?
I am looking for infinite set of Diophantine solutions.
Suppose we require
$$0<\min(a,d)<\max(a,d)<\min(b,c)<\max(b,c)\leq\sqrt 2\min(a,d)$$
$$a,b,c,d\in\mathbb Z$$
then can we still find ...
- 1,826
4
votes
1
answer
332
views
On the Diophantine equation $x^{5} + y^5 = z^p$
Let $x, y, z$ be pairwise coprime positive integers. Does one have $x^5 + y^5 = z^p$ for any prime $p \geq 2$ ?
- 1,019
12
votes
1
answer
832
views
Basic prerequisite (topics) to read current research in Diophantine equation for an independent researcher
I have completed studying Galois theory, Fermat's Last Theorem for Regular prime and some number theoretic complex analysis (prime number theorem), and basic linear forms in logarithm.
What else ...
4
votes
0
answers
300
views
On $\sum_{k=1}^nk^3 = x^3 + y^3$ with $x,y \ge 1$
My question is related to https://oeis.org/A269839.
It is well-known that there are parametric families of solutions for cubes that are sums of consecutive cubes: https://arxiv.org/pdf/1603.08901.pdf. ...
- 701
7
votes
2
answers
1k
views
Is it true that $\{x^3-2x+y^3-2y+z^3-2z: x,y,z\in\mathbb Z\}=\mathbb Z$?
A well known conjecture states that
$$\{x^3+y^3+z^3:\ x,y,z\in\mathbb Z\}=\{m\in\mathbb Z:\ m\not\equiv\pm4\pmod 9\}.$$
For $m=33,\, 42$ an integer solution to the equation $x^3+y^3+z^3=m$ was only ...
- 15.6k
2
votes
3
answers
179
views
Solutions to $(a^c-b^c)+m(r^c-s^c)=0$ in integers
Let $c\ge2$ be a fixed positive integer. How many nontrivial solutions in the integers does the equation $(a^c-b^c)+3(r^c-s^c)=0$ have? If $c=2$, I think it has infinitely many solutions as it seems ...
- 2,089
10
votes
1
answer
2k
views
Is new $n$-conjecture as follows correct?
Given a positive integer $P>1$, let its prime factorization be written as$$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}.$$
Define the functions $h(P)$ by $h(1)=1$ and $h(P)=\min(a_1, a_2,\ldots,...
- 1,776
6
votes
2
answers
713
views
Origin and variations of problem on $4xy-x-y$ being square
One of the forms in which the Diophantine equation in question can be found in the literature is this:
Solve the equation \begin{eqnarray}z^{2} = 4xy-x-y \qquad \qquad (\ast)\end{eqnarray} in ...
- 8,842
10
votes
0
answers
259
views
Integral points on elliptic curve and the Lee norm
This question is based on small experiments I have done in Sagemath and if it is not research level, I will move it to MSE:
Let $E$ be an elliptic curve defined with coefficients in $\mathbb{Z}$.
The ...
user6671
1
vote
1
answer
157
views
Diophantine equations that involve Gregory coefficients: a computational exercise
In this post that I've asked three weeks ago with same title in Mathematics Stack Exchange and identificator 3692235, for integers $k\geq 1$, we denote the Gregory coefficients as $G_k$. Wikipedia has ...
3
votes
2
answers
374
views
Diophantine equation that has an infinite number of positive integers solutions
Let us consider a sequence of continuous functions $g_{q}:ℝ^2\to ℝ^2$. Let $(A_{q})_{q\geq 1}$ be a sequence of compact sets in $ℝ^2$. Assuming that each function $g_{q}$ is topologically mixing in $...
- 1,197
4
votes
0
answers
199
views
Finding integer solutions to $n=a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-ac-bc)$?
Is there anything known how to find integer solutions to
$$n=a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-ac-bc)$$
where $n$ is a natural number $\neq 3,6 \mod 9$ and $a,b,c \in \mathbb{Z}$?
Notice ...
user6671
1
vote
1
answer
330
views
A paper by W. Ljunggren
I am looking for the following paper by Ljunggren, Wilhelm: "Zur Theorie der Gleichung $x^2 + 1 = Dy^4$", Avh. Norske Vid. Akad. Oslo. I., 1942 (5): 27
The main result of this paper which I am ...
- 2,404
1
vote
1
answer
218
views
On a quadratic diophantine equation
Given a quadratic diophantine equation in $\mathbb Z[x,y,z]$ of form
$$ax^2+by^2+cx+dy+ez+f=0$$ are there standard methods to solve for it when $$\|(x^2,y^2,z)\|_\infty\leq e^{1/2}$$
$$\|(a,b,c,d,e,f)...
- 1,826
4
votes
2
answers
452
views
$x^3+x^2y^2+y^3=7$, and solvable families of Diophantine equations
(a) Do there exist integers $x$ and $y$ such that $x^3+x^2y^2+y^3=7$ ?
(b) Is this equation belongs to some family $F$ of equations for which there is a known algorithms for testing if they have an ...
- 7,182
2
votes
1
answer
139
views
Solutions to the Diophantine equation $a^xy+x=c$
Fix positive integers $a,c$ with $a>2$. Is it possible that the Diophantine equation
$$a^xy+x=c$$
has infinitely many solutions (in positive integers $x$ and $y$)?
- 21
4
votes
1
answer
563
views
The number of perfect squares which can occur in an arithmetic progression of length n
This is a similar question to https://math.stackexchange.com/questions/2023399/the-maximum-number-of-perfect-squares-that-can-be-in-an-arithmetic-progression/3693487#3693487
Let f(n) be the maximum ...
- 2,811
0
votes
0
answers
147
views
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 ...
- 1,826
1
vote
0
answers
225
views
Recursively obtained hard Diophantine equation for "Baseless numbers"
An equivalent problem was originally asked on MSE as Does every number base have at least one “Baseless number”?, but did not receive any answers that would help answer the main question about "...
- 611