New answers tagged number-theory
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$a^3 + b^3 + c^3 = 4abc$ has no positive integer solutions
Actually, mimicking the approach to Solve $a^3 + b^3 + c^3 = 6abc$ (which I was not aware of while posting the question) seems to work, details at
https://artofproblemsolving.com/community/...
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What is the identity of this zeta function?
The identity of the Riemann zeta function is known as Hasse's representations for zeta function. That can be shown easily. First, we defined the forward difference operator as follows.
$$
\Delta f(x)=...
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What is the identity of this zeta function?
Through this site, which was also linked to a Japanese blog, I learned that the formula below was guessed by Knoop and proven by Hasse in 1930. Also, this formula is created through Euler's ...
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$a^3 + b^3 + c^3 = 4abc$ has no positive integer solutions
COMMENT.-Here is a criterion that seems easier to me than the two precedents to try to solve this problem. First, $a,b,c$ must be coprime due to the homogeneity of the equation. If two variables have ...
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Weierstrass Form of degree 4 equation
Yes this is an elliptic curve, and yes we can put it in Weierstrass form. An unhelpful simple answer is that computer algebra will achieve this for you (...
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$a^3 + b^3 + c^3 = 4abc$ has no positive integer solutions
well, why not. You mention factoring. As you say, $x^3 + y^3 + z^3 - 3xyz$ factors over the rationals and factors completely over the complexes, by adding in cube roots of $1.$ However, your ...
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$a^3 + b^3 + c^3 = 4abc$ has no positive integer solutions
I may have an approach, but not the finished solution, for this... Without loss of generality, we may order the positive integers
$$a \ge b \ge c$$
Now there exist integers $d, e \ge 0$ such that
$$ a^...
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How to give this sum a bound?
Expanding on the idea from my comment and the attempt by the OP, I will establish $8\times 15 \times \left(\frac{\pi^4}{90}\right)^2$ is an upper bound for the sum, assuming $x,y\neq0$ in the sum.
...
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Given the primes, how many numbers are there?
To state the question more clearly, we are interested in
$$
\#\{ n\in\Bbb N,\, n\text{ odd}\colon n/2^{\Omega(n)} \le x\}
$$
where $\Omega(n)$ is the number of prime factors of $n$ counted with ...
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Can we (almost always) walk from one Gaussian non-prime to another?
This is just an idea on the 'how to make this question rigorous?' part. I don't have an answer to the actual very interesting question.
Define two Gaussian integers $x,y$ as connected if $|x-y|=1$ as ...
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How to check for solutions of non-linear systems over the integers modulo m?
Let's see, lemma, given a prime $p$ and a target $t,$ there always exist $x,y$ such that $x^2 + y^2 \equiv t \pmod p.$ This one uses just a counting/pigeonhole argument
This can always be done for ...
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A question about integrality in polynomial rings
Pick an $\alpha \in B$, and consider $B'$ the subring of $B$ generated by the $x_i$'s and the coefficients of $\alpha$ (denoted $c_1,...,c_m \in \overline{k}$ in no particular order). Then $\alpha$ is ...
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What's the easiest way to prove the Riemann Zeta function has any zeros at all on the critical line?
The easiest way is to refer to the Hardy $Z$ function,
$$ Z(t) = e^{i \theta(t)} \zeta(1/2 + it), $$
where
$$ e^{i \theta(t)} = \pi^{-i t / 2} \frac{\Gamma(1/4 + it/2)}{\lvert \Gamma(1/4 + it/2) \...
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How to prove that$\sum_{n\le x}(\psi(\frac xn)-\vartheta(\frac xn))\Lambda (n)=O (x) $
Most answers here assume
$$\psi(x) - \vartheta(x) = O(x^{1/2}),$$
which I believe is only proven later in Apostol's Analytic Number Theory book. Prior to the exercise mentioned by OP, the author only ...
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Doubt in proof of theorem 8.20 of Apostol Modular functions and Dirichlet series in number theory
This is a famous result that the partial sums $\zeta_{n}(s)$ has zeros even for $\sigma > 1$, and it must be true since many have already read the argument and bought it.
I guess the author of this ...
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The "Euler Product formula" for general multiplicative functions
A lot can be deduced from the following property:
\begin{equation}
\left(1-\frac{1}{x}\right)\left(1-\frac{1}{x-1}\right)\cdots \left(1-\frac{1}{x-k}\right)=1-\frac{k+1}{x}
\end{equation}
Now, we ...
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The average number of prime factors of a number less than or equal to $N$
Yes, indeed.
For each positive integer $m$, we have
\begin{equation} \sum_{n \le x} \omega(n)= x\log(\log x)+Mx+\sum_{k=1}^{m}C_k \frac{x}{(\log x)^k}+O\bigg(\frac{x}{(\log x)^{m+1}}\bigg) \hspace{...
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The integral basis of $\mathbb{Q}(\zeta+\zeta^{-1})$
First, a priori you want to show that every element of $\mathcal{O}_K$ is of the form $\sum_{i=0}^{\frac{p-3}{2}} t_i (\zeta+\zeta^{-1})^i$ with $t_i\in\mathbb{Z}$, and not of the form $t_0+
\sum_{i=1}...
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Finiteness of a generalization of the class number of a number field
It is true that the number of isomorphism classes of full $\mathcal{O}$-modules in $\mathcal{K}^{\textbf{n}}$ is finite. The result is a special case of the Jordan-Zassenhaus Theorem. For future ...
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Solution of $3^{b+1} = 2^{n+2} - 5$ and $3^{b+1} = 2^{n+2} + 1$ in natural numbers
By a theorem of Bennett (2003) the equation $|2^x-3^y|=z$ has at most one solution unless $z\in\{1,5,7,13,23\}$. For these particular values $z\in\{1,5\}$ there are precisely $3$ solutions. For $z\in\{...
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Discriminant of numbers
It is a standard calculation in number theory that $\text{disc}(x^n-1) = (-1)^{n(n+1)/2 + 1}n^n$, and this implies that $\text{disc}([n]_q) = (-1)^{n(n+1)/2 + 1}n^{n-2}$. Let me prove both these ...
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Fermat's last Theorem and elliptic curve cryptography
There is no relationship between the two.
FLT is about homogeneous equations of the form $x^n+y^n=z^n$ over the integers and states that no integer solution exists when $n\geq 3.$ Yes it is stated in ...
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$H(x)$ approximates $\pi(x)$ pretty well. But what are the drawbacks, when compared with Riemann's $R(x)$?
When $z$ is small, there is
$$
e^z=1+z+O(z^2),
$$
so we have
$$
{e^z-1\over e^z+1}={z+O(z^2)\over2+O(z)}=\frac z2+O(z^2).
$$
This indicates that your $f(x)$ (plug in $z=2/\log n$) is
$$
f(x)\approx\...
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Fermat's last Theorem and elliptic curve cryptography
I don't think there is any connection and the temporal link is weak. First, Fermat's last theorem was proven in 1994. Second, elliptic curve cryptography was proposed in 1985, so about 10 years before ...
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votes
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2-adic valuations of $k\cdot 3^n-1$
The best way to understand the situation, for odd $k$ (even $k$ is trivial) is to decompose your expression, to wit:
$(k\cdot3^n)-1=k[(3^n-1)-(k^{-1}-1)]$
where $k^{-1}$ is the reciprocal of $k$ in $2$...
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votes
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A simple question about bounding a sum
First, the condition $\|x\|_2>\gamma^{-1}$ leads to $\frac{1}{\|\gamma x\|_2} <1$, which implies that $\frac{\gamma^2}{\|\gamma x\|_2^2}$ dominates the other terms in the sum. Hence, we only ...
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Ramification for a Galois extension of number fields
You can find a proof in these lecture notes:
https://websites.math.leidenuniv.nl/algebra/ant.pdf
See theorem 4.14, and also theorem 4.10 if you need the relation between the discriminants of $f$ and $...
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Fibonacci and Lucas numbers related identities
By definition
$$
\sum_{i=1}^n H_i = \sum_{i=1}^n L_i +m \sum_{i=1}^n F_i
$$
where $L_{i\ge 1}=1,3,4,7,\ldots$ and $F_{i\ge 1} = 1,1,2,3,5,\ldots$.
We have a well-known partial sum, see https://oeis....
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Detailed Proof of Proposition 2.12 a) from Diamond, Darmon, Taylor, "Fermat's Last Theorem."
You need to show that for all $\sigma \in G_p$, all $n \ge 1$, and all $\ell^n$-th roots $Q$ of $q$ in $\overline{\mathbf{Q}}_p$, we have $\frac{\sigma(Q)}{Q} \in \mu_{\ell^\infty}$. But $\sigma$ is a ...
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Is $\frac{a^{p}-1}{a-1}$ ($p$ prime) square-free?
Converting my comments into an answer:
This is false for every prime $p$, and the smallest counterexample is $\boxed{ \frac{3^2 - 1}{3 - 1} = 2^2 }$.
For any prime $p$, counterexamples can be ...
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Show that no set of nine consecutive integers can be partitioned into two sets where the products of all elements of two sets are equal.
OP observes that the sequence must be composed of integers that have no prime factors greater than $7$. We also know that if the two subsets have an equal product, then the product of all members of ...
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Is it true that $A[2]\cong \varinjlim_i (A_i[2])$ if $A\cong \varinjlim_{i\in I} A_i$?
I think this should always be true. We have a natural map $\lim\limits_{\rightarrow}(A_i[2])\rightarrow(\lim\limits_{\rightarrow}A_i)[2]$ induced by the inclusions $A_i[2]\hookrightarrow A_i$. We can ...
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Is it true that $A[2]\cong \varinjlim_i (A_i[2])$ if $A\cong \varinjlim_{i\in I} A_i$?
This is true if the colimit is filtered. Then $M[2] = \textrm{ker}(M \stackrel{2}{\to} M )$ is the kernel of a canonical map, and kernels commute with filtered colimits for abelian groups.
In other ...
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How to find the $\zeta$ representation of a $L$-series
Another profitable way to think about these problems is via the Euler product, assuming the coefficients of the Dirichlet series are multiplicative. In the example you gave, we have
$$\sum_{n = 1}^\...
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Prove the binary icosahedral group is isomorphic to ${\rm SL}(2,5)$
Your strategy looks fine to me, let's try to make it work. First, working $\bmod \sqrt{5}$ we can invert $2$ and $\tau \equiv 3 \bmod \sqrt{5}$, so we get $\zeta \equiv -1 + i + 3j \bmod \sqrt{5}$. So ...
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Is A276175 integer-only?
Following up on a comment by David E. Speyer (indicating that the $b_n$ sequence arises from the cluster algebra $\mathcal{A}(Q)$ associated to a certain quiver $Q$), I would like to mention that the $...
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Galois Group of $\mathbb Q$ Surjects onto Certain Cyclotomic Extension
This ultimately reduces to finite Galois theory using topology. Let me explain.
Claim 1: let $G$ be a compact group, and $H$ be a profinite group and $f: G \rightarrow H$ be a continuous homomorphism ...
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Is this polynomial irreducible over the rationals?
We can use a bit of Galois theory. Let $\zeta_n = e^{2\pi i/n}$, so $2\cos \frac{2\pi k}{n} = \zeta_n^k + \zeta_n^{-k}$. Since $T_n(\cos(2\pi k/n)) = \cos(2\pi k) = 1$, we have $\cos(2\pi k/n)$ is a ...
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Number theory problem involving linear congruences
To answer as the OP says "involving linear congruences," suppose the three integers are $a$, $b$, and $c$. Then, by the Division Algorithm, using $r$ as the remainder:
\begin{align}
3a &...
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Norm of a simple extension
As commented by @Daniel5803, the mistake was in the calculation of the last column of the matrix, because $\phi(a^{n-1})=xa^{n-1}-a^n$ but $a^n$ is not in your basis.
Let us first prove the special ...
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Nontrivial integer solutions of $ a^3+b^3=c^3+d^3$ and $a+b=c+d$
We cut to the chase.
Given
$a+b=c+d\not=0\tag{1}$
$a^3+b^3=c^3+d^3\not=0\tag{2}$
Divide (2) by (1) to get
$a^2-ab+b^2=c^2-cd+d^2\tag{3}$
but also square (1) to get
$a^2+2ab+b^2=c^2+2cd+d^2\tag{4}$
...
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If the analytic rank is one then the sign in the functional equation is -1?
This is something you can definitely use without a reference. The functional equation is $L(f,s)=\epsilon L(f,2-s)$ for some sign $\epsilon$.
Let $F(z)=L(f,1+z)$, then $F(z)=\epsilon F(-z)$.
So if $\...
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How can I prove my conjecture for the coefficients in $t(x)=\log(1+\exp(x)) $?
The question posed by Gottfried Helms at How can I prove my conjecture for the coefficients in $t(x)=\log(1+\exp(x)) $? is equivalent to the known Maclaurin power series exapsnion
\begin{equation}\...
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How to show isomorphism between an adele ring on a number field and its Pontryagin dual?
(This should probably just be a comment, but I don't have enough reputation yet).
One can reduce this problem to proving that local fields are self-dual by theorem 5.4, which states that the ...
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Calculation of Hilbert symbol over $\Bbb Q_p$ where $p$ is an odd prime
If you read that section it says "over the $p$-adics..." so $u, v$ are not meant to be integers but $p$-adic integers. If $a = \frac{1}{2}$ and $p = 3$ then $\alpha = 0$ and $u = \frac{1}{2}$...
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Question on a Quora Proof of Fermat's Last Theorem for Exponent 3
I believe I have concocted an answer to my question. Suppose instead, without loss of generality, that $\lambda$ divides $x$. Then write
$$y^3 + u(-z)^3 = (-x)^3$$
By an exercise from the post, this ...
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The equation $x^3=u$ has a solution in $\mathbb{Z}_p^{\times}$ if and only if $x^3 \equiv u \mod p$ has a solution.
First, $x^{3} = p$ implies $v_p(x^3) = v_p(u) = 0 \iff v_p(x) = 0$ so $x \in \Bbb Z_{p}^\times$. So we can write $x = a_0 + a_1p + a_2p^2+\cdots$, where $a_{i}$ is an integer between $0$ and $p-1$.
...
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Upper bound on number of divisors of $x^2 -1$
If we define $$f(n):=\frac{\frac{\ln(\sigma_0(n^2-1))}{\ln(n^2-1)}\cdot \ln(ln(x^2-1))}{1.5379 \ln(2)}$$ the above inequality transofrms for an integer $n>1$ to $f(n)<1$. In this sense , the ...
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Question on a Quora Proof of Fermat's Last Theorem for Exponent 3
The proofs I know start as follows. Let $(x,y,z)$ be a nontrivial integer solution to $x^3+y^3=z^3$ with coprime $x,y,z$. Then we have $3\mid xyz$, because otherwise $x^3+y^3\equiv -2,0,2\bmod 9$ and $...
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$\frac{1}{8}x^2+\frac{23}{9}y^2=z^2$ has non-trivial solutions $\in \mathbb{Q}$
and all integer solutions to $2x^2 + 23 y^2 = z^2$ with $\gcd(x,y,z) = 1$ and $x,y,z \geq 0$ come from
$$ x = |u^2 + 10 uv + 2 v^2| \, , \; \; y = |u^2 - 2 v^2| \, , \; \; z = 5 u^2 + 4 uv + 10 ...
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