Questions tagged [number-theory]
Questions on advanced topics - beyond those in typical introductory courses: higher degree algebraic number and function fields, Diophantine equations, geometry of numbers / lattices, quadratic forms, discontinuous groups and automorphic forms, Diophantine approximation, transcendental numbers, elliptic curves and arithmetic algebraic geometry, exponential and character sums, Zeta and L-functions, multiplicative and additive number theory, etc.
41,822 questions
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Is $137 = 2^7 + 3^2$ the unique sum of a power of $2$ and a power of $3$? [closed]
I noticed that $137$ can be written as:
$$137 = 2^7 + 3^2 = 128 + 9.$$
I'm trying to determine: Is this the only way to express $137$ as $2^a + 3^b$ where $a, b$ are positive integers?
Checking ...
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Classification of perfect squares with digits strictly less than 7 such that adding 3 to each digit yields another square
Here is the problem I am investigating:I am looking for all perfect squares $N$ (in base 10) that satisfy two conditions:Digit Constraint: Every digit of $N$ is strictly less than $7$ (i.e., digits $\...
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Galois degree $p$ extensions of $\mathbb{Q}$ that are ramified only at an odd prime $p$
This question is motivated by a question on Constructive Kroneker-Weber asked by user quanta and an answer by user David E Speyer to this question (which limits itself to regular primes). In order to ...
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Explicit bounds for polytope integral
In Theorem 1.1 of this paper a constant is defined by a massive polytope integral, which is subsequently evaluated for certain values of $r$ using a computer algebra system. My question is whether ...
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$2^n-1$ can't be a powerful number?
If $2^n - 1 = a^b$ with $b>1$ , then by Mihailescu's Theorem such $(n, a, b) > 1$ would not exist.
Moreover , can we always find a prime $p$ such that $v_p(2^n-1)=1$ ?
i.e : $2^n-1$ can't be ...
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Let $r_n$ be the least $N$ s.t. every $p^n$-th power mod $\mathfrak{m}_K^N$ lifts. Do we almost have $r_n=r_{n-1}+e$ for $e$ the ramification index?
$\renewcommand{\O}{\mathcal{O}_K}
\newcommand{\m}{\mathfrak{m}_K}$
I'm afraid that this post is much too long, but I do want to present my motivations and ideas below. I would be really sorry if you ...
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Diophantine equation $x^3 + y^3 + z^3 = 2$
It's well known that the Diophantine equation $x^3 + y^3 + z^3 = 2$ has infinitely many solutions of the form $x = -6n^2$, $y = 1 + 6n^3$, $z = 1 - 6n^3.$ Most people working this area appear to avoid ...
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Formulas to find out the critical limit of decomposing numbers
I'm doing a proof on Graph theory on a colorability problem of a connected graph. My proof method involves decomposing numbers into the sum of smaller numbers according to specific rules. In ...
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Prove that the number of sign changes $\sim x^{3/4}$ (square-free integers)
Let $Q(x)$ be the number of square-free integers up to $x$. The asymptotic formula is well known, namely $$Q(x) = \frac{x}{\zeta(2)} + \Delta(x)$$ with error term $\Delta(x)$. It is also known that $\...
user1295974
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A characterization of bases $b$ for which the digit-sum dynamical system has only periods $\{1,2b\}$
Consider the dynamical system on $\mathbb{𝑍}_{\ge 0}^2$
$$T_b(x,y)=(y,s_b(x)+s_b(y))$$
where $s_b(n)$ is the sum of the digits of $n$ in base $b$. For a fixed base $b$, denote by $P(b)$ the set of ...
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Many degree $d$ nilpotent elements of quotients of polynomial rings and non-vanishing product
The question survived bounty on mathoverlfow
Generalization of this question.
Let $n$ be positive integer and $f_1(x_1,...,x_k), \dots, f_m(x_1,...x_k)$
polynomials with integer coefficients.
Let $K=\...
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A divisor identity proven by Liouville
I recently encountered the following identity involving sums and divisors online:
$$\left(\sum_{k|n}d(k)\right)^2 = \sum_{k|n}d(k)^3$$
where $d(n)$ denotes the number of positive integer divisors of $...
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Are the discriminants $-12$, $-16$, $-27$, $-28$ known to have class number $1$?
I have been studying imaginary quadratic fields with class number $1$. The Heegner-Stark theorem states there are $9$ fundamental discriminants with this property: $-3$, $-4$, $-7$, $-8$, $-11$, $-19$,...
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Show that $p_{3n} \nmid n^2+1$
If $p_n$ denotes the $n$th prime, show that $p_{3n} \nmid n^2+1$.
If $p_{3n} \mid n^2+1$ it means that $-1$ is a quadratic residue modulo $p_{3n}$ and so $p_{3n} = 1 \pmod{4}$. How do we continue ...
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Radium Rabbit Conjecture, version 3.0: The fractional part of the square of the area of a triangle with odd-integer sides is $\frac{3}{16}$. [closed]
Why Version 3.0?
In the earlier versions of this conjecture, I focused on triangles whose side lengths are distinct prime numbers.
Through the discussion that followed, it became clear that the ...
