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6 votes
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Efficient solutions to general Bézout’s identity $a_1 b_1 + \dots + a_n b_n = 1$

Suppose I have integers $a_1, \dots, a_n$ which are coprime, meaning that $$a_1 b_1 + \dots + a_n b_n = 1$$ has a solution in integers $b_1, \dots, b_n$. I would like to get a bound saying ...
Kim's user avatar
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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 ...
mathoverflowUser's user avatar
4 votes
0 answers
332 views

Diophantine equations over cyclotomic fields

Let $\mathbb{Q}^{\text{ab}}$ be the compositum of all finite abelian extensions of $\mathbb{Q}$. Explicitly, $\mathbb{Q}^{\text{ab}}$ is the field obtained from $\mathbb{Q}$ by adjoining all roots of ...
Pablo's user avatar
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3 votes
0 answers
148 views

Solutions of a quadratic Diophantine equation over algebraic integers

The problem is not very exactly Diophantine in the classical way. I am trying to find some algebraic integer solutions in a number ring to a quadratic equation over the same ring. Precisely, let $\...
Yanlong Hao's user avatar
2 votes
0 answers
115 views

Reference request: "A result of Siegel" related to Ramanujan-Nagell type equations

Wikipedia refers to the Diophantine equation $ x^2 + D = AB^n $ as an "equation of Ramanujan–Nagell type". It also says that "A result of Siegel implies that the number of solutions in ...
Eric Nathan Stucky's user avatar
2 votes
0 answers
125 views

How many integers below $n$ can be expressed as a sum of $k$ $m$th powers?

For $m,k \geq 2$, let $C_{m,k}(n)$ denote the number of positive integers less than or equal to $n$ which can be expressed as a sum of $k$ $m$th powers. I am interested in the asymptotic behavior of ...
aras's user avatar
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2 votes
0 answers
207 views

n-ary quadratic forms with $S$-integer values

Let $Q(x_1,\ldots,x_n):=x_1^2+\cdots+x_n^2$ be an $n$-ary quadratic form. Given a finite set of (rational) primes $S$ is there an algorithm or theorem that describes all solutions to $Q(x_1,\ldots,...
Eric Rowell's user avatar
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2 votes
0 answers
243 views

Hurwitz integers and $F_4$

The Hurwitz integers are $$ \mathcal H= \{a+bi+cj+dk:a,b,c,d\in\mathbb Z\;\text{ or } \;a,b,c,d\in \tfrac12+\mathbb Z\}. $$ I want to know if there is a formula, for $m\in\mathbb Z$, for the number ...
emiliocba's user avatar
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1 vote
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Mahler's proof of $S$-unit equation

Many modern proofs of the (ineffective) finiteness of solutions of the $S$-unit equation $x+y=1$ use Roth's theorem. In particular it is used Lang's version of Roth's theorem which takes in account ...
manifold's user avatar
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1 vote
0 answers
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Beyond pure rational and integral solutions to cubic equations

I started reading Silverman and Tate’s introductory book on elliptic curves. In the introductory chapter they mention that for the Bachet equation $x^2 - y^3 = c$, there are infinitely many rational ...
John Jiang's user avatar
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120 views

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 ...
Safwane's user avatar
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138 views

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 ...
user142929's user avatar
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0 answers
180 views

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 ...
user avatar
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133 views

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 ...
user142929's user avatar