All Questions
Tagged with reference-request nt.number-theory
388 questions with no upvoted or accepted answers
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Construction of RM abelian variety from eigenform
Let $f$ be a normalized eigenform of weight $2$ level $N$. If the Fourier coefficients of $f$ generate a totally real field $F$, then we associate to $f$ a system of $\ell$-adic Galois representations ...
- 15.4k
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140
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State of the art on attempts to solve the elliptic curve discrete logarithm problem through transfering the problem to a weaker curve
Let an elliptic curve $E$, and 2 points on such curve $P$ and $O$ the methods I’m talking about consist in creating a weaker elliptic curve $F$ and mapping $P$ and $O$ to $F$ while successfully ...
- 87
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101
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Identities for Prime Coefficients of Certain Cusp Forms
While working with Fourier expansions of cusp forms of congruence subgroups of the modular group, I observed the following patterns in their prime coefficients.
Let $a(n)$ be the Fourier coefficients ...
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108
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looking for reference for two elliptic curves with equal formal group
I am looking for a reference.
In this post, @Chris Wurthrich made the following comment:
If the formal group laws (probably upon particular choice of coordinates) of two elliptic curves over any ring ...
- 195
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81
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Computing elliptic periods from modular form
How are the periods of a modular elliptic curve computed as path integrals of its associated normalized weight 2 cusp form on the modular curve? Please provide specific paths for both periods and cite ...
- 2,907
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75
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Existence of smooth integers in every residue class with large modulus
Let us say that a positive integer $x$ is $y$-power smooth, if the largest prime power divisor of $x$ is at most $y$. In what follows, let $C$ be any real number larger than $1$ and, for an integer $x$...
- 1,663
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73
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Decrease of $(1/\zeta)^{(r)}(\sigma + i T)$ as $\sigma\to -\infty$?
What is a standard reference for the simple fact that, for $T$ fixed and $\sigma\to -\infty$,
every derivative $|(1/\zeta)^{(r)}(\sigma+i T)|$ of the Riemann zeta function decreases faster than any ...
- 20.2k
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200
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Kato's explicit reciprocity law paper
Does anyone have a copy of Kato's article Generalized explicit reciprocity laws in Advanced Studies in Contemp. Math which is used heavily in his paper constructing his eponymous Euler system? I used ...
- 2,044
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115
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Conjecture that there are finitely many integer powers $a^x$ and $b^y$ such that $b^y - a^x = n$: who first came up with it?
I came up with an interesting mathematical conjecture: for every natural number $n$ there is only a finite number of integer powers $a^x$ and $b^y$ such that $b^y - a^x = n$.
I would like to find out ...
- 161
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121
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Quadratic residue problem involving prime divisors of a polynomial
Let $n$ be a square-free natural number, and let $f\in\mathbb{Z}[x]$ be monic and irreducible of degree $\geq2$. I am trying to determine whether there always exists a prime $p$, $p\nmid n$, ...
- 1
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196
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Sum of squares squared in an arithmetic progression
Let $r(n)$ be the number of ways to write $n$ as a sum of two squares and $(a,q)=1$.
What is known about
$$
\sum_{n \le x,n \equiv a (\text{mod} \, q)} r(n)^2 \quad?
$$
I am looking for uniform ...
- 130
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157
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On the mean value of Dirichlet L-function
Could you please provide a link to the source?
$$\sum_{\chi\neq \chi_0}\int_{0}^{T}|L(1/2+it,\chi)|^4dt\ll (qT)^{1+\varepsilon},$$ where $\chi_0$ is the principal character modulo $q$, and $L(s,\chi)$ ...
- 150
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171
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Total sum of characters over partitions with distinct parts
In my earlier quest, we looked at $\chi_{\mu}^{\lambda}=$value of an irreducible character of the symmetric group $\frak{S}_n$, where $\mu$ and $\lambda$ are (unrestricted) partitions of $n$. Then, ...
- 43.2k
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138
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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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A way to bound $\sum_{1 \leq n \leq X} \min ( \| \alpha n \|^{-1} , X/n)$?
Let $\alpha$ be a real number and $|| \cdot ||$ be the distance to
the nearest integer.
I want to find a non-trivial upper bound for
$$
\sum_{1 \leq n \leq X} \min ( || \alpha n ||^{-1} , X/n),
$$
...
- 3,625
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87
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Reference request for additive persistence of a number
It is well known fact that each natural number can be represented uniquely in any base. So we can define digit sum function whose value is sum of digits of the natural number in given base.
Let $f(n,b)...
- 249
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91
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Image of Frobenius element under irreducible representation is diagonalizable
Let $K/ \mathbb Q$ be a Galois extension, and $\rho$ be an irreducible representation of the Galois group $Gal(K/ \mathbb Q)$. Consider an integer prime $p$ which doesn't ramify in $K$, and let $\...
- 739
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177
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Status of the $n$ conjecture and, as secondary question or reference request, what about a transfer method for this conjecture $n>3$
The n conjecture is a generalization of the abc conjecture. What is the current status of the $n$ conjecture? See also [1]
Question 1. Can you tell us what about the current status of the $n$ ...
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226
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On Prime Numbers which can be Norms of an Integral Ideal of a Number Field
We know that since the ring $\mathbb Z [i]$ of Gaussian integers is a Principal Ideal Domain, the only integer primes which can norms of some ideal of $\mathbb Z [i]$ are those which can be expressed ...
- 739
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154
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On the equation that involves the Dedekind psi function $\psi(x)=n$ with unique solution $x$, for a fixed integer $n\geq 1$
The Dedekind psi function is defined for a positive integer $m>1$ as
$$\psi(m)=m\prod_{\substack{p\mid m\\p\text{ prime}}}\left(1+\frac{1}{p}\right)\tag{1}$$
with the definition $\psi(1)=1$. See ...
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133
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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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83
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Is it possible to get a conjecture similar to Mandl's conjecture for a different arithmetic function of number theory, mainly related to primes?
I'm curious to know if are in the literature analogous conjectures to the conjecture due to Mandl, I ask about these analogous conjectures for different sequences playing an important role in number ...
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86
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Polynomials of integer coefficients that evaluated at Mersenne or Fermat numbers produce square-free integers
Mersenne numbers $M_n=2^n-1$ and Fermat numbers $F_n=2^{2^n}+1$ draw the attention of professional mathematicians to get prime constellations or statements related to primality tests for these ...
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122
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References for the extension of Euler's phi function to number rings
Can anyone post a self-contained reference concerning the extension of the Euler phi function to number rings and its basic properties (reminiscent of those that the classic Euler phi function has)? ...
- 123
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132
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Final step in Coppersmith?
In the final step in Coppersmith technique we have $n$ polynomials (possibly non-homogeneous) in $\mathbb Z[x_1,\dots,x_m]$ where $m\leq n$ and using elimination theory we extract the common integer ...
- 13.9k
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759
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On sets of coprime integers in intervals
Briefly,
Question: Is it "good enough" to use least prime factor in choosing a maximal set of coprime integers in an interval?
The post title comes from a 1993 paper of Erdos and Sarkozy. They ...
- 13k
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116
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Reference request for bounds of $n$-th composite
Motivation
I will briefly elaborate here my motivations for asking the question. If you are not interested in it then please go to the questions.
Recently during trying to understand and prove the ...
user57432
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115
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What is known about the lower bound for the integers $n$ for which $n$ minus the first $k$ odd primes are $k$ composite numbers?
Question edited in view of the comments below
By Yamada's paper we can conclude that if $n>e^{e^{36}}$ be an even number then it can always be written as the sum of a prime and a semi-prime.
My ...
user57432
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142
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Mobius function on values of an irreducible quadratic polynomial
Are there infinitely many integers $n$ for which $n^2 + 1$ is square-free, and has an even number of (necessarily distinct) prime factors ?
- 11.3k
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257
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Hercules and the Hydra with time constraints
The game of Hercules vs. the Hydra can be put in terms of a single number in hereditarily-factorized form. For example, if the Hydra is $2^{19^3} \cdot 5^{11^7}$, Hercules must choose between two ...
- 2,302
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98
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Eigenvalues of a sequence of matrices involving the divisor function
Let $A_{n,k},k=1,\ldots,n$ be a sequence of $n\times n$ upper triangular matrices where $A_{n,1}=I_n$ and $A_{n,k},\quad 2\leq k\leq n$ be a regularly shifted and scaled matrix, with $P_{n,k}$ an $n\...
- 10.4k
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234
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whether the quotient of continued fraction of algebraic irrational number is bounded or not is similar or equivalent to Collatz conjecture?
I vaguely recall that whether the quotient of continued fraction of algebraic irrational number is bounded or not is similar or equivalent to Collatz conjecture, could any one give the reference? or ...
- 3,094
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266
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Completing a dyadic sum
Suppose I knew the behaviour of a given sum in every other interval, for example:
$$
\sum_{\substack{0\leq a \leq x\\ a\equiv 1 (k)}} \sum_{x/(a+k/2)< b \leq x/a} f(b) \sim g(x),
$$
for any $x>1$...
- 3,799
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197
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'Adelic torus' not arising from a rational torus
Let $G$ be a reductive group over a global field $F$, and $\gamma$ a strongly regular semi-simple element of $G(F)$. Then the centralizer $G_\gamma$ is defines an $F$-torus $T$, and hence by base ...
- 3,799
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Bound of Chebyshev function and zeros of zeta function
It is an elementary argument (such as in Multiplicative Number Theory, section 18) that, if the Chebyshev's function $f(x) = \sum_{n \le x} \Lambda(x) = x + O(x^\alpha)$ for some $\alpha < 1$, then ...
- 101
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234
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On the irrationality measure of generalized Stoneham numbers
Pick non-zero integers $a,b,c$ with $a,b \ge 2$ and let $\xi_{a,b,c}$ be the sum of the series $\sum_{n=1}^\infty a^{-b^n} c^{-n}$ (no restriction is made on the sign of $c$); when $b = c$ and $\gcd(a,...
- 10.5k
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"Descending cohomology, geometrically" by Mazur:
(Exist texts of that talk or related texts: http://ttv.mit.edu/collections/harris60/videos/13881-problem-session-barry-mazur ?) Article: http://www.math.harvard.edu/~mazur/papers/page37.pdf
- 10.8k
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215
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Invariance of the (Liouville-Roth) irrationality measure under rational Möbius transformations
For a real number $x$, we define the (Liouville-Roth) irrationality measure of $x$, here denoted by $\mu(x)$, as the infimum, with respect to the poset $(\mathbb{R}_0^+ \cup \{\infty\}, \le)$, of the (...
- 10.5k