All Questions
Tagged with nt.number-theory diophantine-equations
787 questions
175
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2
answers
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Estimating the size of solutions of a diophantine equation
A. Is there natural numbers $a,b,c$ such that $\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b}$ is equal to an odd natural number ?
(I do not know any such numbers).
B. Suppose that $\frac{a}{b+c} + \...
- 1,881
72
votes
3
answers
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Can you solve the listed smallest open Diophantine equations?
In 2018, Zidane asked What is the smallest unsolved Diophantine equation? The suggested way to measure size is substitute 2 instead of all variables, absolute values instead of all coefficients, and ...
- 7,182
63
votes
11
answers
8k
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Why certain diophantine equations are interesting (and others are not) ?
It is quite clear why certain differential equations, among the jungle of possible diff equations that is possible to conceive, are studied: some come from physical problems, or from "spontaneous" ...
- 23.3k
58
votes
3
answers
4k
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What is the geometry of an undecidable diophantine equation?
As an arithmetic algebraic geometer of the highest moral fiber, I am trained to look at Diophantine equations in terms of the geometry of the corresponding scheme. For instance, if the Diophantine ...
- 148k
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votes
7
answers
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What is the smallest unsolved Diophantine equation?
If $P=\sum_{\bf i} a_{\bf i}x^{\bf i}\in {\mathbb Z}[x_1,\dots,x_d]$, let $|P|=\sum_{\bf i}|a_{\bf i}|x^{\bf i}$ and $h(P)=|P|(2,\dots,2)$, so that there is only a finite number of $P$ with $h(P)\leq ...
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votes
4
answers
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Why do Pell equations appear in Ramanujan's pi formulas?
While answering this MSE question about the Pell equation $x^2-29y^2=1$, I noticed that certain fundamental solutions appeared in Ramanujan's famous pi formula.
I. Given the fundamental unit $\...
- 12.6k
48
votes
4
answers
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Fermat's last theorem over larger fields
Fermat's last theorem implies that the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}$ is finite.
Is the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}^{\text{ab}}$ finite?
Here $\...
- 11.3k
44
votes
1
answer
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Infinitely many solutions of a diophantine equation
If $P(x,y,...,z)$ is a polynomial with integer coefficients then every integer solution of $P=0$ corresponds to a homomorphism from $\mathbb{Z}[x,y,...,z]/(P)$ to $\mathbb{Z}$. So there are infinitely ...
user6976
43
votes
3
answers
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Which integers can be expressed as a sum of three cubes in infinitely many ways?
For fixed $n \in \mathbb{N}$ consider integer solutions to
$$x^3+y^3+z^3=n \qquad (1) $$
If $n$ is a cube or twice a cube, identities exist.
Elkies suggests no other polynomial identities are known.
...
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40
votes
1
answer
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Diophantine equation for 2016: triangular $|{\rm GL}_2({\bf F}_q)|$
For a prime power $q$ the group ${\rm GL}_2({\bf F}_q)$ has
$(q^2-1)(q^2-q)$ elements. This happens to be a triangular number for
$q=2$ (being $6 = 1+2+3$), and $-$ more notably, especially this year ...
- 79.9k
40
votes
2
answers
3k
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$x^4+y^4$ powerful for relatively prime $x,y$
I asked this question on the NMBRTHRY mailing list on
17 February 2014, but it remains unsolved as far as I know.
Recall that a "powerful
number" is a positive integer whose prime ...
- 79.9k
38
votes
5
answers
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Are nontrivial integer solutions known for $x^3+y^3+z^3=3$?
The Diophantine equation
$$x^3+y^3+z^3=3$$
has four easy integer solutions: $(1,1,1)$ and the three permutations of $(4,4,-5)$. Elsenhans and Jahnel wrote in 2007 that these were all the solutions ...
- 2,350
36
votes
1
answer
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On a remark of Tait on FLT for the exponent 3
This is one of those recreational questions that aren't really about research. I found a curious remark in an old volume of American Mathematical Monthly (1922) which I'll quote below:
In the ...
- 85.6k
35
votes
3
answers
5k
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Solve in positive integers: $n!=m(m+1)$
Does anybody know a solution to this problem? (Sorry, I've missed one summand in the previous post.)
- 12.3k
35
votes
3
answers
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A binomial generalization of the FLT: Bombieri's Napkin Problem
This is an extract from Apéry's biography
(which some of the people have already enjoyed in
this answer).
During a mathematician's dinner in
Kingston, Canada, in 1979, the
conversation turned ...
- 13.5k
34
votes
1
answer
843
views
Can we write each positive rational number as $\frac1{p_1-1}+\ldots+\frac1{p_k-1}$ with $p_1,\ldots,p_k$ distinct primes?
It is well-known that any positive rational number can be written as the sum of finitely many distinct unit fractions. This is easy since
$$\frac1n=\frac1{n+1}+\frac1{n(n+1)}\quad\text{for all}\ n=1,2,...
- 15.6k
33
votes
1
answer
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About the validity of a new conjecture about a diophantine equation
Let us consider the following conjecture:
Conjecture: There are no integer solutions of the equation $$x^{y-z}z^{x-y}=y^{x-z}$$ with $x,y,z$ distinct positive integers greater than or equal to $2$.
...
- 1,197
33
votes
4
answers
4k
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Can the difference of two distinct Fibonacci numbers be a square infinitely often?
Can the difference of two distinct Fibonacci numbers be a square infinitely often?
There are few solutions with indices $<10^{4}$ the largest two being $F_{14}-F_{13}=12^2$ and $F_{13}-F_{11}=12^2$...
- 25.4k
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votes
1
answer
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Are there any integers which can't be written as a sum of two fourth powers minus a cube?
To be precise, I am asking:
Does there exist an integer $k$ such that there do not exist (possibly negative) integers $x,y,z$ satisfying $x^4+y^4=z^3+k$?
Heuristically the answer must be yes, in ...
- 8,688
31
votes
5
answers
8k
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Fermat's proof for $x^3-y^2=2$
Fermat proved that $x^3-y^2=2$ has only one solution $(x,y)=(3,5)$.
After some search, I only found proofs using factorization over the ring $Z[\sqrt{-2}]$.
My question is:
Is this Fermat's original ...
- 3,866
31
votes
1
answer
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Can $9xy$ divide $1+x^2+x^3+y^2$?
Can $9xy$ divide $1+x^2+x^3+y^2$ for integers $x,y$? Equivalently, do there exist integers $x,y,z$ such that
$$
1 + x^2 + x^3 + y^2 + 9 x y z = 0 \quad ?
$$
This equation arises in the search for the ...
- 7,182
31
votes
3
answers
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Trost's Discriminant Trick
The following little trick was introduced by E. Trost
(Eine Bemerkung zur Diophantischen Analysis,
Elem. Math. 26 (1971), 60-61). For showing that a diophantine equation
such as $x^4 - 2y^2 = 1$ ...
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30
votes
9
answers
10k
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Diophantine equation with no integer solutions, but with solutions modulo every integer
It's probably common knowledge that there are Diophantine equations which do not admit any solutions in the integers, but which admit solutions modulo $n$ for every $n$. This fact is stated, for ...
- 10.3k
30
votes
2
answers
1k
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Is equation $xy(x+y)=7z^2+1$ solvable in integers?
Do there exist integers $x,y,z$ such that
$$
xy(x+y)=7z^2 + 1 ?
$$
The motivation is simple. Together with Aubrey de Grey, we developed a computer program that incorporates all standard methods we ...
- 7,182
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votes
5
answers
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Parametric solutions of Pell's equation
Given a positive integer $n$ which is not a perfect square, it is well-known that
Pell's equation $a^2 - nb^2 = 1$ is always solvable in non-zero integers $a$ and $b$.
Question: Let $n$ be a ...
- 19.6k
28
votes
6
answers
2k
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Patterns among integer-distance points
Mark each point of $\mathbb{N}^2$ ($\mathbb{N}$ the natural numbers) if its
Euclidean distance from the origin is an integer. One obtains a plot like this, symmetric about the $45^\circ$ diagonal.
...
- 151k
27
votes
4
answers
11k
views
Is there an elementary way to find the integer solutions to $x^2-y^3=1$?
I gave this problem to my undergraduate assistant, as I saw that Euler had originally solved it (although I am having trouble finding his proof). After working on it for two weeks, we boiled the hard ...
- 18.7k
27
votes
3
answers
1k
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When does $axy+byz+czx$ represent all integers?
For which $a,b,c$ does $axy+byz+czx$ represent all integers?
In a recent answer, I conjectured that this holds whenever $\gcd(a,b,c)=1$, and I hope someone will know. I also conjectured that $axy+byz+...
user44143
27
votes
2
answers
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Fermat's Last Theorem in the cyclotomic integers.
Kummer proved that there are no non-trivial solutions to the Fermat equation FLT(n): $x^n + y^n = z^n$ with $n > 2$ natural and $x,y,z$ elements of a regular cyclotomic ring of integers $K$.
I am ...
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answer
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Polynomials with rational coefficients
Long time ago there was a question
on whether a polynomial bijection $\mathbb Q^2\to\mathbb Q$ exists. Only one attempt
of answering it has been given, highly downvoted by the way. But this answer isn'...
- 13.5k
27
votes
1
answer
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Solutions to $\binom{n}{5} = 2 \binom{m}{5}$
In Finite Mathematics by Lial et al. (10th ed.), problem 8.3.34 says:
On National Public Radio, the Weekend Edition program posed the
following probability problem: Given a certain number of ...
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votes
1
answer
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The "stubborn" solutions to sums of three cubes
It is conjectured (see [1]) that for any integer $k\not\equiv \pm 4\pmod 9$ there are infinitely many integer solutions to
$$
a^3+b^3+c^3=k.
$$
Numerical investigations of this conjecture show that ...
- 7,193
26
votes
1
answer
943
views
What are the integer solutions to $x^4+2y^4=3z^4$?
This equation has the obvious integer solution $(x,y,z)=(\pm 1,\pm 1,\pm 1)$. By Faltings's theorem, the equation has finitely many primitive integer solutions (those with $\gcd(x,y,z)=1$).
What is ...
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3
answers
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The diophantine eq. $x^4 +y^4 +1=z^2$
This question is an exact duplicate of the question
Does the equation $x^4+y^4+1=z^2$ have a non-trivial solution?
posted by Tito Piezas III on math.stackexchange.com.
The background of ...
user14413
26
votes
2
answers
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Proving non-existence of solutions to $3^n-2^m=t$ without using congruences
I made a passing comment under Max Alekseyev's cute answer to this question and Pete Clark suggested I raise it explicitly as a different question. I cannot give any motivation for it however---it was ...
- 41.4k
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votes
2
answers
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Why do the $2$-Selmer ranks of $y^2 = x^3 + p^3 $ and $y^2 = x^3 - p^3 $ agree?
I was playing around with sage, when I found that the Mordell-Weil ranks (over $\mathbb{Q}$) of the elliptic curves $y^2=x^3+p^3$ and $y^2=x^3-p^3 $ almost always agree, for $p$ prime. The first few ...
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Pythagorean 5-tuples
What is the solution of the equation $x^2+y^2+z^2+t^2=w^2$ in polynomials over C ("Pythagorean 5-tuples")?
There are simple formulas describing Pythagorean n-tuples for n=3,4,6:
n=3. The formula ...
- 1,488
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votes
2
answers
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Are (55, 165, 495, 1485) and (286, 1716, 10296, 61776) the only geometric sequences of length 4 among non-trivial binomials?
Let's define non-trivial binomial coefficients as values of $\binom{n}{k}$, where $n$ and $k$ are positive integers such that $2 \le k \le \frac{n}{2}$. (Therefore, $6$ is the smallest non-trivial ...
- 425
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votes
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answer
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Algorithmic (un-)solvability of diophantine equations of given degree with given number of variables
Question: For which $d, k \in \mathbb{N}$ is there an algorithm to determine
whether a polynomial diophantine equation
$$
P(x_1, \dots, x_k) = 0, \ \ \ P \in \mathbb{Z}[x_1, \dots, x_k]
$$
...
- 19.6k
24
votes
2
answers
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On the smallest open Diophantine equations: beyond Hilbert's 10 problem
In 2018, Zidane asked What is the smallest unsolved Diophantine equation? The suggested way to measure size of the equation is substitute 2 instead of all variables, absolute values instead of all ...
- 7,182
23
votes
1
answer
2k
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Ramanujan's pi formulas with a twist
Given the binomial function $\binom{n}{k}$.
1. Define the following sequences,
$$\begin{aligned}
u_1(k) &= \tbinom{2k}{k}\tbinom{3k}{k}\tbinom{6k}{3k} = 1, 120, 83160, 81681600,\dots \\
u_2(k) &...
- 12.6k
22
votes
9
answers
21k
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Is there an algorithm to solve quadratic Diophantine equations?
I was asked two questions related to Diophantine equations.
Can one find all integer triplets $(x,y,z)$ satisfying $x^2 + x = y^2 + y + z^2 + z$? I mean some kind of parametrization which gives all ...
- 223
22
votes
4
answers
6k
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The modular arithmetic contradiction trick for Diophantine equations
It is a slick, and seemingly ad-hoc, technique often used to prove that a Diophantine equation has no solutions.
The equation $f(x_1,\ldots, x_k)=0$, with variables $x_i\in\mathbb{Z}$ and some ...
- 2,075
22
votes
2
answers
7k
views
Which types of Diophantine equations are solvable?
Is there a list somewhere of which types of Diophantine equations are solvable, which types are not solvable, and which types are not known to be solvable or not? (When I say solvable, I mean that we ...
- 2,649
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votes
2
answers
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Why is 1331 the only cube of the form $x^2 + x − 1$?
The Wikipedia (https://en.wikipedia.org/wiki/1000_(number)#1300_to_1399) mentions that 1331 is the only cube of the form $x^2 + x − 1$, for $x = 36$. What is the proof?
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votes
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answer
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Permutations $\pi\in S_n$ with $\sum_{k=1}^n\frac1{k+\pi(k)}=1$
Let $S_n$ be the symmetric group of all the permutations of $\{1,\ldots,n\}$.
Motivated by Question 315568 (http://mathoverflow.net/questions/315568), here I pose the following question.
QUESTION: ...
- 15.6k
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votes
2
answers
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State of knowledge of $a^n+b^n=c^n+d^n$ vs. $a^n+b^n+c^n=d^n+e^n+f^n$
As far as I understand, both of the Diophantine equations
$$a^5 + b^5 = c^5 + d^5$$
and
$$a^6 + b^6 = c^6 + d^6$$
have no known nontrivial solutions, but
$$24^5 + 28^5 + 67^5 = 3^5+64^5+62^5$$
and
$$3^...
- 151k
21
votes
1
answer
740
views
Does $A-A=\mathbb Q$ hold for $A=\{x^4+y^4:\ x,y\in\mathbb Q\}$?
Let $A=\{x^4+y^4:\ x,y\in\mathbb Q\}$. Then
$$A-A:=\{a-b:\ a,b\in A\}=\{u^4+v^4-x^4-y^4:\ u,v,x,y\in\mathbb Q\}.$$
Motivated by Question 415482, here I ask the following question.
Question. Is it true ...
- 15.6k
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votes
1
answer
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Is "almost-solvability" of Diophantine equations decidable?
Say that a Diophantine equation is almost-satisfiable iff for each $n\in\mathbb{N}$ it has a solution mod $n$. Trivially genuine satisfiability over $\mathbb{N}$ implies almost-satisfiability, but the ...
- 21.2k
21
votes
3
answers
872
views
Consecutive square values of cubic polynomials
Let $P(x)$ be a cubic polynomial with integer coefficients. Does there exist a constant $c$ such that at least one of the following values $P(0),P(1),...,P(c)$ is not a square?
It is known that the ...
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