Questions tagged [diophantine-equations]
Diophantine equations are polynomial equations $F=0$, or systems of polynomial equations $F_1=\ldots=F_k=0$, where $F,F_1,\ldots,F_k$ are polynomials in either $\mathbb{Z}[X_1,\ldots,X_n]$ of $\mathbb{Q}[X_1,\ldots,X_n]$ of which it is asked to find solutions over $\mathbb{Z}$ or $\mathbb{Q}$. Topics: Pell equations, quadratic forms, elliptic curves, abelian varieties, hyperelliptic curves, Thue equations, normic forms, K3 surfaces ...
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Natural number solutions for equations of the form $\frac{a^2}{a^2-1} \cdot \frac{b^2}{b^2-1} = \frac{c^2}{c^2-1}$
Consider the equation $$\frac{a^2}{a^2-1} \cdot \frac{b^2}{b^2-1} = \frac{c^2}{c^2-1}.$$
Of course, there are solutions to this like $(a,b,c) = (9,8,6)$.
Is there any known approximation for the ...
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On a structural decomposition of polynomials based on integral roots
Given an irreducible polynomial of structure $$f(x,y)=\sum_{\substack{i,j\in\{0,1,2\}\\i+j\leq3}}a_{i, j}x^iy^j\in\mathbb Z[x,y]$$ with $a_{2,1}a_{1,2}a_{1,1}a_{1,0}a_{0,1}a_{0,0}\neq0$ if $f(m,n)=0$ ...
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When is $\phi(a^n+b^n+c^n)=0\mod n$?
A corollary Zsigmondy's Theorem leads to the following congruence (one can look to $(24)$),$\phi(a^n+b^n)=0\mod n$ whenever $a, b$ are coprime and $n \neq 2$ and $(a,b)\neq(1,1)$. (Here $\phi$ is the ...
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$3$-ranks of elliptic curves and representations $p=ax^3+by^3$
Let $p$ be a prime with $p\equiv2\pmod3$ and $E_p$ the elliptic curve $y^2=x^3+9p^2$
which has a rational $3$-torsion point. Let $\alpha$ from $E_p(\mathbb Q)$ to $\mathbb Q^*/{\mathbb Q^*}^3$ be the $...
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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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Small linear relations between primitive Pythagorean triples $\mathsf{II}$
WillJagy answered a linear relation question on Pythagorean Triples in Small linear relations between primitive Pythagorean triples $\mathsf I$.
Now let $a^2+b^2=c^2$ be a primitive Pythagorean ...
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Cohn's eight diophantine equations
Today I was reading J.H.E. Cohn's Eight diophantine equations (1966). On p. 158
he comes across the equation $y^2 = a^3 + 3a$ for odd values of $a$ and writes that this is equivalent to $x^3 + (x+1)^3 ...
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FLT and integral points on elliptic curves
For integers $x,y,z,t,n$ define $S_n : xy(x+y)=t^n$.
For $ n > 2$, Fermat's Last Theorem implies there are no integral
solution on $S_n$ with $x,y$ coprime and $xy(x+y) \ne 0$ since $x,y,x+y$ are
...
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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
$$|\{(n_1,n_2,\ldots,n_k)\in\mathbb{Z}^k: \ n_1^2+n_2^2+\cdots+n_k^2=N\}|\leq C_\varepsilon N^{\frac{...
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Write $n^2$ as $x^2+y^2+2\times4^z$ or $x^2+y^2+5\times 4^z$
In March 2018, I formulated the following somewhat curious question.
Question 1. Whether for any integer $n>1$ there is a nonnegative integer $k$ such that $n^2-2\times 4^k$ or $n^2-5\times 4^k$ ...
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$n$-variable polynomials modulo $p$
The Hasse-Weil bound implies that for any 2-variable polynomial $P(x,y)$, there exists approximately $p$ solutions in $\mathbb{F}_p$ of $P(x,y) \equiv a \pmod p$ for sufficiently large $p$, and any ...
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Write $p^2$ as $x^2+2y^2+3\times 2^z$ with $x,y,z$ nonnegative integers
In April 2018, I noted that the first integer $n>1$ with $n^2\not\in\{x^2+2y^2+3\times 2^z:\ x,y,z=0,1,2,\ldots\}$ is $$5884015571=7\times17\times49445509.$$
Question. Is it true that for each ...
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Diophantine equation $10^n-a^3-b^3=c^2$
Consider the Diophantine equation:
$10^n-a^3-b^3=c^2$, for $a$, $b$, $c$, and $n$ positive.
Has this equation infinitely many solutions?
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Simplest diophantine equation with open solvability
What is the simplest diophantine equation for which we (collectively) don't know whether it has any solutions? I'm aware of many simple ones where we don't know (whether we know) all the solutions, ...
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Solutions in primes of the equation $\,3p^2+q^2=r^2+3$
Let's consider the Diophantine equation $\,3p^2+q^2=r^2+3$.
Actually, I am interested only in the solutions represented by sets $\,(p,q,r)\,$ of prime numbers.
It's easy to prove that if $\,(p,q)\,$ ...
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Does this equation have more than one integer solution?
Consider the following diophantine equation
$$n = (3^x - 2^x)/(2^y - 3^x),$$ where $x$ and $y$ are positive integers and $2^y > 3^x$.
Does $n$ have any other integer solutions besides the case ...
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On Kellner's result and the Erdos-Moser equation
Let $m, k$ be positive integers. Consider the Erdos-Moser expression $S_{k}(m) = 1^k + 2^k + ... + (m-1)^k$. By a result of Kellner, we know that if $m | S_{k}(m)$, then $m|B_k$, where $B_r$ denotes ...
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Which Hilbert's 10th polynomials are known to have solutions?
The Diophantine equation
$$x^3 + y^3 + z^3 = 42$$
was recently solved by
Booker and Sutherland:
Sum of three cubes for 42 finally solved.
Is there a clean partition of the form of those
polynomial ...
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Request for an exact formula related to a partition in number theory
The Frobenius equation is the Diophantine equation $$
a_1 x_1+\dots+a_n x_n=b,$$
where the $a_j$ are positive integers, $b$ is an integer, and a solution $$(x_1, \dots, x_n)$$
must consist of non-...
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Around the diophantine equation $\frac{a}{2b+3c}+\frac{b}{2c+3a}+\frac{c}{2a+3b}=\text{odd integer}$, over positive integers
I am interested to know if a similar theorem that shows this answer of the post
Estimating the size of solutions of a diophantine equation (this MathOverflow, January 5th 2016) is feasible for a ...
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find all of rational solutions of the quartic equation?
Consider the equation $a^4+6v^2a^2-8a+v^4=0$ over the rationals.
Note that the following are solutions: $(a,v)=(1,1),(0,0),(2,0)$. Are there any other rational solutions?
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On variants of the abc conjecture in terms of Lehmer means
In this post we denote the Lehmer mean of a tuple $\text{x}$ of positive real numbers as $$L_p(\text{x})={\sum_{k=1}^nx_k^p\over\sum_{k=1}^nx_k^{p-1}},$$
see the reference Wikipedia Lehmer mean.
The ...
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Sum from combinatorics on nonnegative integer numbers
Let $n_1,n_2,\ldots,n_k\in\{0,1,2,\ldots\}$. Can you calculate the sum
$$
\sum_{n_1,n_2,\ldots,n_k\geqslant0}\mathbb{1}_\left\{n_1+\frac{n_2}{2}+\ldots+\frac{n_k}{k}<1\right\}?
$$
If it's helpful, ...
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Diophantine representation of the set of prime numbers of the form $n²+1$
A polynomial formula for the primes (with 26 variables) was presented by Jones, J., Sato, D., Wada, H. and Wiens, D. (1976). Diophantine representation of the set of prime numbers. American ...
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Integer points of one Mordell equation
How can I determine all integer points of the following equation
$$y^2=x^3+10546$$
I tried Magma with
IntegralPoints(EllipticCurve([0,10546]));
but got the ...
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Research work on $ax^n-by^m=1$
I am looking for results on the equation $$ax^n-by^m=1 \tag 1 $$ where $\gcd(m,n)=1$ and $a,b,n,m$ are constants.
I found literature for $ax^n-by^n=1$ (R. A. Mollin, D. T. Walker) but couldn't ...
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Prove $\frac{\text{Area}_1}{c_1^2}+\frac{\text{Area}_2}{c_2^2}\neq \frac{\text{Area}_3}{c_3^2}$ for all primitive Pythagorean triples
A while ago I asked this question on MSE here. After placing a bounty it got quite a bit of attention but unfortunately it has yet to be resolved. After getting some advice from MO Meta I have decided ...
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A question about integer triples
How can we generate all integer solutions of the equation
$$(qr+rp+pq)(x^2+y^2+z^2) = (p^2+q^2+r^2)(yz+zx+xy),$$
given that $p,q,r$ are integers?
Clearly if any one of $(x,y,z), (x,z,y), (y,z,x), (...
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On sparse $0/1$ linear equations solvable with compressed sensing
If you have a system of $m$ linearly independent equations in $n$ variables with domain $0/1$ and we know there is at least one solution with at most $d$ variables to be $1$ then if $m$ at least a ...
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Diophantine equations that involve cubes and the volume of square frustums
This week I wondered about diophantine problems that involve the volume of certain cubes and frustums, see the Wikipedia Frustum. I wondered if each one of these problems have infinitely many ...
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Solutions of a Diophantine equation with large common divisors
There is a curious Diphantine equation showing up in my research:
$$ \frac{1}{a^2-1}+\frac{1}{b^2-1}=\frac{1}{c^2-1}+\frac{1}{d^2-1}. $$
I am trying to find its integer solutions where $a$, $b$, $c$ ...
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Different solution of power Diophantine equation based on constant term
Let us define a power Diophantine equation by 2 algebraic functions $f,g$ (having different degree) and by integers $k, l >0$ where, there are finite solutions for $f(x)+k=g(y)$, but there exists $...
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Diophantine equation in Laurent polynomials
(This is a modified repost of a question from MSE; since it came out of research, I thought it might be appropriate to post it here.)
Consider the equation
\begin{equation*}
P(x, x^{-1})^m + Q(x, x^{-...
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Quadratic factors of $l_1(x,y)^3+l_2(x,y)^3+l_3(x,y)^3-n$
Related to sum of three squares and this question.
Let $l_1,l_2,l_3 \in \mathbb{Z}[x,y]$ and $n \in \mathbb{Z}$.
Assume that $n$ is not a cube and not twice cube.
Let $f=l_1(x,y)^3+l_2(x,y)^3+l_3(x,...
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Rational Diophantine set for the non-squares
Related to Hilbert's Tenth problem.
Is there polynomial with integer coefficients $P(a,x_1,...,x_n)$
such that $P(A,X_i)=0$ has rational solutions $X_i$ iff $A$ is
not the square of integer (or as ...
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On a variant of Brocard's problem using the definition of Pochhammer symbols
I've considered the following variant of Brocard's problem $$\frac{(2n-1)!}{(n-1)!}+1=m^2\tag{1}$$
for integers $n\geq 1$ and integers $m\geq 1$. I was inspired from the fact that the evaluation of ...
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Is integer circuit membership undecidable?
According to wikipedia
integer circuit
in its simplest form is succinct representation of multivariate polynomial with
integer coefficients. Decidability if an integer is represented by the integer ...
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Points on hyperelliptic curves: $y^2=5(x^2-3)(x^2+2)(x^2-11/5)$
González-Jiménez and Xarles studied a problem in Diophantine number theory and they obtained several nice results via elliptic curve Chabauty's method over quadratic number fields. At page 73 in paper ...
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Catalan numbers, Pochhammer symbols, Stirling numbers of the second kind, and sums of aliquot parts
For integers $N\geq 1$ we define $$s(N)=\sigma(N)-N$$ the aliquot sum function, where $\sigma(N)=\sum_{1\leq d|N}d$ is the sum of divisors function.
Here $(x)_n$ is the Pochhammer symbol and ${a\...
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On the number of solutions of the equation involving Pochhammer symbols $(n)_a\cdot(n)_b=(n)_c$, for integers greater than or equal to $2$
As paticular case of the equation involving Pochhammer symbols $$(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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Equations involving arithmetic functions of primorials
Let $\sigma(n)=\sum_{1\leq d\mid n}d$ the sum of divisors, $\varphi(n)$ the Euler's totient function and we denote the primorial $\prod_{k=1}^n p_k$ as $N_n$, where $p_k$ denotes the $k$-th prime ...
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Special linear Diophantine system - is it solvable in general?
Background: An equivalent question was asked on MSE almost two years before this post now. It was never fully resolved. - Here, we are asking if further progress can be made.
Motivation
Solving this ...
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Near Pochhammer symbols: the equation $(n)_m-(k)_l=2$ for integers greater than or equal to two
In this post I consider the following equation involving Pochhammer symbols,
$$(n)_m-(k)_l=2\tag{1}$$
for positive integers $n\geq 2$ and $k\geq 2$, and positive integers $m\geq 2$ and $l\geq 2$.
...
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How small can $u$ be in the Pell equation $u^2-k^3 v^2=\pm 1$?
Let $k$ be positive integer, not a square and let $u_k,v_k$ be non-trivial
solutions to the Pell equation $u_k^2-k^3 v_k^2=\pm 1$.
Q1 How small $u_k$ can be infinitely often as function $k$?
This ...
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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
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Concise formulation of set of equation systems
I have the following set of equation systems, and I would like to find a short, formal way to write it down. My main difficulty is that I cannot find a good way to write the indices of the variables $\...
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On Richard Guy's problem D5 in "Unsolved problems in Number Theory"
This question is motivated by recent discovery of Andrew Booker+Andrew Sutherland.
Richard Guy's problem D5 in his Unsolved Problems in Number Theory contains the original question for the sum of ...
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On the equation involving Stirling numbers of the second kind ${n\brace a}{m\brace b}={k\brace c}$, and its solutions satisfying certain requirements
In this post we denote the Stirling numbers of the second kind as ${r\brace s}$ and we consider the proposal to ask if the equation of the title has infinitely many solutions $${n\brace a}{m\brace b}={...
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Solubility of a Diophantine equation [closed]
Is it true that for positive integers $B$ and $G$, the following equation in $B_1,B_2,G_1,G_2$ has a non-negative integer solution for any value $c$ such that $|c|\le B G$ :
$$G_1B_1-G_2B_2 = c$$
...
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What are the solutions of this Diophantine equation?
Besides $(x, y, z)=(0, 0, 0)$ and $(1, 1, -2)$ (and their permutations) are there any other integer solutions to the equation
$$3(x^{3}+y^{3}+z^{3})+3(x^{2}+y^{2}+z^{2})+(x+y+z)=0 $$ ?
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