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Tagged with diophantine-equations reference-request
62 questions
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Reference request: on sums of the form $ax^m + by^n = h$
I know that equations of the form
$$\displaystyle ax^d + by^d = h$$
with $a,b,h \in \mathbb{Z}$ have been thoroughly investigated as a special (and interesting) case of the Thue-Mahler equation, for ...
- 26.9k
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Papers related to a diophantine equations about Magic square of squares for $n=3$
The open problem of magic squares of squares explained here. Consider the following magic square of squares:
$$
\begin{aligned}
&a^2&b^2&&c^2\\\\
&d^2&e^2&&f^2\\\\
&...
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Different solution of power Diophantine equation based on constant term
Let us define a power Diophantine equation by 2 algebraic functions $f,g$ (having different degree) and by integers $k, l >0$ where, there are finite solutions for $f(x)+k=g(y)$, but there exists $...
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Reference request: Markoff type equations
Consider the equation $x^2+ay^2+bz^2=(1+a+b)xyz.$ If there are infinitely many integral solutions, then up to permutations $(a,b)=(1,1),(1,2),(2,3).$ I have found a presentation by Waldschmidt (https:/...
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Upper bounds for solutions to a Pell-like equation
Let $N$ be a fixed positive integer that is not a square and $m$ be any nonzero integer. Let $x$ and $y$ be positive integers that solve $$x^2 - N y^2 = m^2$$ with $x + y$ minimal (in light of the ...
- 2,283
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Erdős-Straus with 4 terms
The Erdős-Straus conjecture states that any fraction of the form $\frac{4}{n}$ can be decomposed as an Egyptian fraction with just 3 terms. In related research, I've recently come across conditions on ...
- 2,235
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The strong twin conjecture can be transformed into the unsolvability of a particular Diophantine equation
Let us consider the strong twin conjecture:
For all positive integer $n$ there exist a prime $p$ such that
$$n+4<p<2^n2^4$$ and $p$ is a prime and $p+2$ is a prime
Since the inequalities and the ...
- 1,197
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A diophantine equation involving partial sums of exponentials similar than the equation in Fermat's Last Theorem
I'm curious about the following diophantine equation from my invention: I don't know if this is in the literature, I wrote it using creativity in an attempt to write a variant of the equation in ...
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When is $\phi(a^n+b^n+c^n)=0\mod n$?
A corollary Zsigmondy's Theorem leads to the following congruence (one can look to $(24)$),$\phi(a^n+b^n)=0\mod n$ whenever $a, b$ are coprime and $n \neq 2$ and $(a,b)\neq(1,1)$. (Here $\phi$ is the ...
user147204
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What about an alternative formulation for different prime constellations in the spirit of Suzuki's theorem for twin primes?
It is known that the twin prime conjecture is a special case of the $k$-tuple conjecture. See if you want the article with title k-Tuple Conjecture from the encyclopedia Wolfram MathWorld.
On the ...
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Is the conjecture $min(A,B) \le rad(ABC)$ new and correct? [closed]
$\DeclareMathOperator\rad{rad}$Conjecture: If $A, B, C$ are positive integers with $\gcd(A, B)=1$, $\gcd(B, C)=1$, and $\gcd(C, A)=1$, and if $A+B=C$, then $\min(A,B) \le \rad(ABC)$.
If the ...
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Has nontrivial solution in positive integers of a diophantine equation: $x_1^2+x_2^2+x_3^2+x_4^2=y_1^2+y_2^2+y_3^2+y_4^2$ [closed]
Has nontrivial solution in positive integers of a diophantine equation as follows ?
$$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 ...
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