# 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 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.

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### Theorem 1.27 a) in Diamond, Darmon, Taylor, "Fermat's Last Theorem"

I am looking for a reference for part a) of Theorem 1.27 here regarding the proof of the growth of coefficients of cusp forms. Theorem 1.24 gives a very sketchy argument of a less specific fact, but I ...
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### Problem about Elliptic regulator on a set of points in an elliptic curve

I am a bit on the following problem: Let $\{P_1,\ldots,P_r\}$ be a set of rational points on an elliptic curve $E/\mathbb{Q}$ and $\mathfrak{H}:=(\langle P_i,P_j\rangle)_{1\le i,j,\le r}$. Then, I ...
1 vote
39 views

### Similarity of Lifting the Exponent Lemma in Pell numbers

Pell number is a term of the sequence $\{P_n\}$ determined by a recurrence relation $$P_{n+2}=2P_{n+1}+P_n, P_0=0, P_1=1.$$ Let $v_p(x)$ be the $p$-adic valuation of an integer $x$ (the number of ...
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1 vote
29 views

### Help me solve this diophantine equation- $x^5+x^4+1=7^y$ [duplicate]

An obvious soloution is $(x,y) = (2,2)$, but I can't figure out which module should I look for to show there are no more solutions...
• 11
1 vote
51 views

### What is the equation for the number of integer solutions of $n$ for $n^2\equiv -1\pmod m$ where $0≤n<m$, given $m$?

I can’t work out how to find the equation for the number of integer solutions of $n$ for $n^2\equiv -1\pmod m$ for any $m$ where $0≤n < m$. The number of solutions, when non-zero, seems to be ...
24 views

### What Irreducible cubic curve and Siegel's theorem

Let $c(x, y)$ be a cubic polynomial with rational coefficients which is irreducible over $\mathbb{Q}$. I'm trying to understand when the equation $c(x) = 0$ has infinitely many integral points. If $c$ ...
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### Order of Odd Elements in $\mathbb{Z}_{2^n}$

I'm wondering if there is a way to calculate the order of odd numbers in the cyclic ring $\mathbb{Z}_{2^n}$. I found a paper that shows that for any odd $x$, $x^{2^{n-2}} \equiv 1 \mod{2^n}$, and that ...
71 views

### What is a sufficient and necessary condition for a system of linear equations to have a unique integer solution?

Bézout's theorem states that a linear diophantine equation $ax+by=c$ has integer solutions if and only if $\gcd(a,b)|c$. Then is there a theorem regarding the existence of a unique integer solution ...
1 vote
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### Example of the real numbers $x$ appearing in the Jarnik's theorem: $||nx||\leq n^{-\beta}$ for infinitely many $n\in\mathbb{N}$ for fixed $\beta>1$

According to Falconer (Falconer, '85, The Geometry of Fractal Sets, theorem 8.16, p. 134), a part of the Jarnik's theorem is the following: Take $\beta>1$. The set of real numbers $x$ for which ...
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1 vote
24 views

### Generators of coprime ideals perfect powers

Could you please help with this a practice exam problem?: Let $K=\mathbb{Q}(\sqrt{-22})$ with unit group $\mathcal{O}_k^\times=\{ \pm 1 \}.$ Suppose that $x,y,z\in\mathcal{O}_K$ are non-zero, satisfy ...
• 11
1 vote
26 views

### Subspace of newforms one-dimensional with CM $\implies$ unique newform a Poincare series.

Let $k \geq 2$. Say $S_{k}^{\text{new}}(\Gamma_{0}(N), \chi)$ is one-dimensional and spanned by a newform with CM, and $S_{k}^{\text{old}}(\Gamma_{0}(N), \chi)$ has positive dimension. Must it be true ...
• 1,675
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### Period of Lehmer sequences

In his thesis (1930), D. Lehmer did not provide the general formula for the period of his sequences. And it does not appear in HC. Williams book about E. Lucas work. And, since Lehmer sequences are ...
• 303
46 views

### Orthogonal group of Lorentzian lattice $I_{1, n}$ is infinite for $n\geq 2$

I am looking for a reference or an elementary proof of the following fact: The orthogonal group of the lattice $I_{1,2} = I^+ \oplus 2 I^-$ is ifinite. Here the Lorentzian lattice $I_{1,n}$ is given ...
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### Elliptic curves of the form $y^2=x^3+p^2$, $p$ a prime, has torsion subgroup isomorphic to $\mathbb{Z}/3\mathbb{Z}$.

Suppose that $p\ge 5$ is a prime and $P$ be a torsion point on the elliptic curve $E_5:y^2=x^3+p^2$. Using the Nagell-Lutz theorem we can write down all possible values of $y(P)$ (where $P=(x(P),y(P))$...
70 views

### Write $7 \cdot 10^{100} + 7$ as a sum of four squares

How do you write $7 \cdot 10^{100} +7$ as a sum of four squares? I know that you can write it as a sum of four squares by the Lagrange's Four Squares Theorem, but I don't know how to write such a big ...
1 vote
120 views

### An unusual equivalent form of Riemann hypothesis

Let $G(x)=\sum_{k\leq x}\frac{\mu(k)}{k}$, where $\mu$ is the Mobius function. From this question and its answer, its mention the Riemann hypothesis is equivalent to $G(x)=O(x^{-\frac{1}{2}+\epsilon})$...
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1 vote
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### How we can show $\Lambda (s,\chi) < \exp(C|s| \log |s|)$ as $|s| \rightarrow \infty$?

We know that the functional equation for Riemann-Zeta is $\psi(s)=\frac{1}{2}s(s-1)\pi^{-1/2s}\Gamma(1/2s) \zeta(s)$ and $\psi(s) < \exp(C|s| \log |s|)$ as $|s| \rightarrow \infty$. Does it make ...
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### Eigenvalues of matrices corresponding to imaginary part of non-trivial Riemann zeroes.

I have an amateur interest in mathematics and in 2020 after being inspired by a Video on $\zeta$ function by Grant Sanderson, and doing further reading into topics like Hilbert-polya conjecture I ...
1 vote
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### when the fractional ideal $S^{-1}I$ of the localization $S^{-1}A$ of a dedekind domain $A$ is principal [closed]
if a fractional ideal $S^{-1}I$ of the localization $S^{-1}A$ of a dedekind domain $A$ is principal, is that implies conclusion about the fractional ideal $I$ in $A$