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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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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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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 ...
joro's user avatar
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Mathematical Aspects of Hectoc-type Puzzles

hectoc is a puzzle, where one is given a sequence of six decimal digits and the task is to intersperse arithmetic operations from the given set $+,-,/,*$ and matching brackets $(,)$ in a way that the ...
Manfred Weis's user avatar
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Rhythmic identities, equations, domains and frustrations

A rhythmic identity: Let $\ \mathbb K\ $ be a field of characteristics $\ne 2\ \ (i.e.\ 1+1\ne 0).\ $ Then $$ \forall_{x\in\mathbb K}\quad \binom{\binom x2}2+\binom{\binom {x+1}2}2\ \ =\ \ ...
Włodzimierz Holsztyński's user avatar
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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 ...
Włodzimierz Holsztyński's user avatar
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Systems of linear modular equations with unknowns in the moduli

I am interested in systems of linear modular equations, where the unknowns also appear in the moduli. The general form would be: $A \vec{x}= \vec{b} \;\textrm{mod} \; (C \vec{x}+\vec{d})$ where A ...
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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}$...
Jason Sawyer's user avatar
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Using the circle method to prove that there are no solutions to diophantine equaltions

Would it be possible to use the circle method to prove that there are no solutions to certain diophantine equations. For example, could one use the circle method to prove the fact that there are no ...
Mayank Pandey's user avatar
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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. ...
Turbo's user avatar
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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?...
joro's user avatar
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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 $X^2-PY^2=k$ ...
Jason Smith's user avatar
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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.
Raghav Bhutani's user avatar
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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 $n^2-5\times 4^k$ ...
Zhi-Wei Sun's user avatar
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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 $x^3 +y^3 = z^4$ ?
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A curious Diophantine problem

Let $a, b, c, d$ be positive integers where $\gcd(a, b)=\gcd(c, d)=\gcd(b, d)=\gcd(bd, ad+bc)=1$ and $\min(b, d)>1$. Is it possible to have $$bd(ad+bc)^{2}\varphi(a)\varphi(c)=ac\varphi(bd)\varphi^{...
Q_p's user avatar
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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 ...
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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 ...
Zhi-Wei Sun's user avatar
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1 answer
149 views

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 ...
math110's user avatar
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1 answer
96 views

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?
Enzo Creti's user avatar
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1 answer
301 views

List of obscure summation identities [closed]

I am trying to evaluate a fairly simple summation: $\sum_{k=1}^n ka^kb^{n-k}$ Which is related to the common identity for $\sum_{k=1}^n ka^k$ available on Wikipedia. I've previously seen lengthy lists ...
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1 answer
201 views

How many integer points does my favorite ellipse goes through? [closed]

What value on P gives an ellipse with 768 lattice Points? x^2 + 3y^2 = P P= 4*7*13*19*31*37*43 gives 384 lattice points
Sture Sjöstedt's user avatar
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260 views

Is the conjecture $min(A,B) \le rad(ABC)$ new and correct? [closed]

$\DeclareMathOperator\rad{rad}$Conjecture: If $A, B, C$ are positive integers with $\gcd(A, B)=1$, $\gcd(B, C)=1$, and $\gcd(C, A)=1$, and if $A+B=C$, then $\min(A,B) \le \rad(ABC)$. If the ...
Đào Thanh Oai's user avatar
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2 answers
148 views

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
sonu's user avatar
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1 answer
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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-(...
zeraoulia rafik's user avatar
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1 answer
168 views

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?
Enzo Creti's user avatar
-2 votes
1 answer
180 views

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 ...
mojojojo's user avatar
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4 answers
230 views

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 $\chi(v)=0$ ...
JMP's user avatar
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1 answer
201 views

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 $(mn,pq)=1$, $\alpha_i\in\Bbb Z$ ...
Turbo's user avatar
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-3 votes
1 answer
193 views

$p=4x^2+27y^2$,with $p$ a prime [closed]

p is a prime ,on what condition the Diophantine equation is solvable.what is it Linear expression ,for example ,$x^2+3y^2=p$, $p=3k+1$ ,$x^2+5y^2=p$ , $p=1,9\pmod{20}$.
wsc's user avatar
  • 13
-3 votes
2 answers
160 views

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 ...
Benjamin L. Warren's user avatar
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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 ...
Q_p's user avatar
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2 answers
272 views

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 ...
Cố Gắng Lên's user avatar
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1 answer
201 views

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 $$ x^k+y^k-z^k-w^k=3\quad \ (x,y,z,w\in\mathbb N=\{0,1,2,\ldots\})\tag{1}$$ ...
Zhi-Wei Sun's user avatar
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2 answers
233 views

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?
Mike's user avatar
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1 answer
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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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