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Results tagged with nt.number-theory
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user 48142
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
7
votes
Divisibility condition implies $a_1=\dotsb=a_k$?
Here's a a tweak of Seva's idea that gives a counterexample. Note that if $r$ is odd, then $2^{n}+1$ divides $2^{rn} + 1$.
Let $k = 6$, $a_{1} = 1$, $a_{2} = a_{3} = a_{4} = 2$, $a_{5} = a_{6} = 4$. T …
- 19.9k
7
votes
Accepted
Isomorphism between genus 1 modular curves and elliptic curves
The maps $j : X_{0}(N) \to \mathbb{P}^{1}$ are given in Magma's ''Small modular curves'' database. In each case, they construct functions on the modular curves $X_{0}(N)$ out of eta products, modular …
- 19.9k
4
votes
Accepted
Generating DataSet of Strong PseudoPrimes?
I'm guessing you will want to be working with numbers larger than 64-bits, and so you probably want GMP (see this page). This library is used by much of the software that number theorists use. (Magma …
- 19.9k
7
votes
Accepted
Question about iterations not divisible by infinitely many prime numbers
Yes. This follows from a result of Corrales-Rodrigáñez and Schoof (see the paper here) solving the support problem of Erdős.
In particular, suppose that there are only finitely many primes $p$ that do …
- 19.9k
12
votes
Accepted
Etale cohomology approach on $\tau(n)$
One of the goals of the development etale cohomology was to generate a cohomology theory that could successfully count points on varieties over finite fields, with one of the main goals of proving the …
- 19.9k
6
votes
Approximations to $\pi$
This is not exactly an answer to the stated question, but it's too long for a comment. Rather than the form given in the question, one could represent a number in the form $\frac{a + b \sqrt{d}}{c}$, …
- 19.9k
4
votes
Accepted
Best error terms for functions related to square free numbers
As I say in the comment, the asymptotics for $M_{+}$ and $M_{-}$ follow directly from those for $M$ and $\hat{M}$. Therefore $M_{+}(x) = \frac{1}{2 \zeta(2)} x + \frac{1}{2} M(x) + O(x^{1/2})$ and $M_ …
- 19.9k
34
votes
Accepted
Question on a generalisation of a theorem by Euler
I suspect that $k = 4$ is good, but am not sure how to prove it. However, every positive integer $k \geq 5$ is good. This follows from the fact (see the proof of Theorem 1 from this preprint) that for …
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22
votes
Accepted
On Fibonacci numbers that are also highly composite
The largest highly composite Fibonacci number is $F_{3} = 2$.
If $p$ is a prime number, then either $p \mid F_{p-1}$ (if $p \equiv \pm 1 \pmod{5}$), $p \mid F_{p}$ (if $p = 5$), or $p \mid F_{p+1}$ (i …
- 19.9k
10
votes
Accepted
Prime numbers in a sparse set
Yes, there is a $c > 1$ for which infinitely many numbers of the form $\lfloor k^{c} \rfloor$ are prime. The first result of this type was proven in Ilya Piatetski-Shapiro's Ph.D. thesis (written in 1 …
- 19.9k
17
votes
Accepted
A divisor sum congruence for 8n+6
The congruence you state is true for all $m \equiv 6 \pmod{8}$. The proof I give below relies on the theory of modular forms. First, observe that
$$
\sum_{k=1}^{m-1} d(k) d(m-k) = 2 \sum_{k=1}^{\frac{ …
- 19.9k
9
votes
Accepted
Complexity of computing the number of visible points
There is an algorithm for computing $F(N) = \# \{ (a,b) : 1 \leq a, b \leq N, \gcd(a,b) = 1 \}$ in time $O(N^{5/6 + \epsilon})$. This relies on the algorithm of Deleglise and Rivat (see their paper he …
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7
votes
The Chebotarev Density Theorem and the representation of infinitely many numbers by forms
The example of Heath-Brown's article is a good one. For a bit more elementary examples, you can fix a number field $K$ with $[K : \mathbb{Q}] = n$ and ring of integers $\mathcal{O}_{K}$ and pick a bas …
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3
votes
Minimum number of unit fractions to sum up a given positive rational
No such polynomial-time algorithm exists because in some instances it would take too long to write down (or store in memory) the answer. In particular, if $n$ is a positive integer, then we have $\sum …
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10
votes
Accepted
Extension of a formula for the quadratic Gauss sums
No, the relation is not as simple in this case. For example, if $k = 3$ and $p = 7$, the three different cubic Gauss sums are roots of $y^{3} - 21y - 7$, and the three roots of this polynomial do not …
- 19.9k
8
votes
$2^n$-1 consisting only of small factors
It is true that if $N > 60$, then $2^{N} - 1$ has a prime factor $> 2500$.
Here's another approach. First, observe that every prime factor of $2^{p} - 1$ is $\equiv 1 \pmod{p}$. Combining this with t …
- 19.9k
3
votes
Accepted
When are the powers of 2 sum-free mod n?
This question is very similar to the one here, and the heuristic should apply equally well. In particular, $A$ is sum-free if and only if there does not exist a $k$ with $k \ne \frac{n+1}{2}$ so that …
- 19.9k
7
votes
Accepted
primitive prime divisor of $2^{8n+4} - 1 $
No. We have that $p = 709$ is a primitive prime divisor of $2^{708} - 1$. However, $\frac{2^{708} - 1}{2^{177} + 1}$ is a multiple of the prime $q = 5521693$ and therefore $q-1 | \gamma\left(\frac{2^{ …
- 19.9k
5
votes
Accepted
$p$-th root of non-torsion points on elliptic curves
No. Lemma 3.7 on page 11 from my paper here implies that if $\mathcal{T}_{1} = {\rm Gal}(K(E[p])/K)$ and there is a normal subgroup $H \unlhd \mathcal{T}_{1}$ with order coprime to $p$ for which $E[p] …
- 19.9k
16
votes
Diophantine equation $3(a^4+a^2b^2+b^4)+(c^4+c^2d^2+d^4)=3(a^2+b^2)(c^2+d^2)$
The equation you specify defines a surface $X$ in $\mathbb{P}^{3}$, and this surface is a K3 surface. It is conjectured that if $X$ is a K3 surface, there is a field extension $K/\mathbb{Q}$ over whic …
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9
votes
Accepted
Congruences among primes modulo which a given polynomial has roots
Here's a survey of the possible things that can happen. In regards to your first question, given any polynomial $f(x)$, there is a positive integer $M$ so that if $\gcd(b,M) = 1$, then there are infin …
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9
votes
Accepted
Sets of squares representing all squares up to $n^2$
We can have the size of $S$ as small as $c \ln(n)$ for some constant $c$, and we can do this in such a way that every element of $\{1, 2, \ldots, n^2 \}$ can be represented by adding or subtracting at …
- 19.9k
4
votes
Relations of eisenstein series with eta quotient
The answer is probably some fairly basic linear algebra. For the first one, each term on the right hand side is a modular form of weight $4$ and level $2$. The space $M_{4}(\Gamma_{0}(2))$ has dimensi …
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8
votes
Accepted
Is $p$ is square modulo $F_p$ when $p=4k+1 > 5$?
The answer to the first question is yes, although the argument I give below is not along the lines that you were originally thinking. I will show that $p$ is a square modulo $q$ for every prime factor …
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40
votes
Accepted
When $\frac{a^2}{b+c}＋\frac{b^2}{a+c}＋\frac{c^2}{a+b}$ is integer and $a,b,c$ are coprime na...
Yes, there is another solution. The next one I found is a bit big, namely
$$ a = 15349474555424019, b = 35633837601183731, c = 105699057106239769. $$
This solution also satisfies the property that
$$ …
- 19.9k
12
votes
Accepted
Extension of $\mathbb Q$ which splits only at primes in $S$
For many choices of $R$ and $S$ the answer is obviously no. For example, if $R$ is empty, then the answer is no, because there are no unramified extensions of $\mathbb{Q}$.
For a more interesting exa …
- 19.9k
4
votes
Accepted
Solutions to diophantine equation
I probably put a little bit too much effort into this. The only rational point on this curve is $(0,0)$ (as well as the points at infinity $(1 : 5 : 0)$ and $(1 : -5 : 0)$).
There's a slightly non-ob …
- 19.9k
10
votes
Accepted
Cusp forms with integer Fourier-coefficients
No. Take $k = 24$ and $N = 1$. Then $\Delta^{2} = q^{2} - 48q^{3} + 1080q^{4} + \cdots \in S_{K}(\Gamma_{1}(N),\mathbb{Z})$. However, if we write $\Delta^{2} = c_{1} f_{1} + c_{2} f_{2}$, where $f_{1} …
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5
votes
Accepted
Representation of integers by positive definite ternary quadratic polynomials with linear terms
Yes, it is possible to extend these methods, although the picture is somewhat
less clear than in the quadratic form case. When one discusses representations by a quadratic polynomial, this is equival …
- 19.9k
6
votes
Accepted
How does this sequence grow
The answer is yes, and the number of solutions with a prime $p$ is $\lfloor \frac{p+5}{8} \rfloor$ when $p \not\equiv 1 \pmod{8}$ and is $\lfloor \frac{p+5}{8} \rfloor + 1$ when $p \equiv 1 \pmod{8}$. …
- 19.9k
7
votes
Accepted
Growth order of numbers whose prime factors are all congruent to +1 or -1 modulo 8
This is an old question (and according to this MO question, the result you seek was proven by Landau). In particular, it follows from this that if $S$ is a set of arithmetic progressions containing a …
- 19.9k
21
votes
Is there a real nonintegral number $x >1$ such that $\lfloor x^n \rfloor$ is a square intege...
At the request of the OP, I am turning my comment into an answer. It is possible to have $\lfloor x^{n} \rfloor$ close to a square for all positive integers $n$. For example, if $x = \frac{7 + 3 \sqrt …
- 19.9k
38
votes
Accepted
When does a Catalan number equal a Fibonacci number?
A result of the type you seek follows easily from Carmichael's theorem, that if $m > 12$, then there is a prime $p$ that divides $F_{m}$, but does not divide $F_{k}$ for $k < m$.
Suppose $C_{n} = \b …
- 19.9k
5
votes
Accepted
Why is this function a modular function of level $5$?
Here's a fairly straightforward way to show that $\phi$ is modular of level $5$ using Siegel functions.
Claim: The function $f(\tau)$ is a modular function for $\Gamma(5)$ if and only if $f(5\tau)$ is …
- 19.9k
2
votes
Accepted
Imaginary quadratic fields with $\ell$-indivisible class number
Here's an elementary argument. For $\ell < 41$, $K = \mathbb{Q}(\sqrt{-163})$ works. For $\ell = 41$, $K = \mathbb{Q}(\sqrt{-3})$ works. Assume then that $\ell \geq 43$.
Choose an integer $1 \leq n \l …
- 19.9k
8
votes
Accepted
An old conjecture of M.Newman
First, I think your definition of $H_{n}$ does not agree with Newman's definition. Newman says the following: "Let $H_{n} \subset G_{n}$ be the set of functions of $G_{n}$ with non-negative valence at …
- 19.9k
32
votes
Only odd primes?
If $k$ is odd and not a perfect square, then the sets are disjoint. In particular, if $\alpha = \frac{k - \sqrt{k}}{\frac{k-1}{2}}$ and $\beta = \frac{k + \sqrt{k}}{\frac{k-1}{2}}$, then $\alpha$ and …
- 19.9k
7
votes
Can the generalized divisor summatory function $D_z$ be expressed explicitly in terms of Zet...
The short answer is that, in all likelihood, a formula of the type you seek only exists if $z$ is a negative integer.
The way to approach a question like this is to first note that $\sum_{n=1}^{\inf …
- 19.9k
2
votes
Accepted
Small Galois group solution to Fermat quintic
I have answers to your first two questions, and some insight into the third.
First, there is a quartic in the family above with Galois group contained in $A_{4}$. One example is found by taking $q = 1 …
- 19.9k
16
votes
Accepted
Prove that $1$ is the sum of three tetrahedral numbers infinitely many different ways
There are infinitely many solutions. I'll show below that there are infinitely many positive integers $k$ for which $93k^{2} - 288k + 276 = z^{2}$ for some positive integer $z$. From such a $z$, we ge …
- 19.9k
8
votes
Examples of models for modular curves
Here's an example. Let's take $\Gamma = \Gamma_{0}(4)$, and
$\Gamma' = \Gamma(2)$. We'll let $\alpha = \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}$, so $\Gamma' = \alpha \Gamma \alpha^{-1}$. The func …
- 19.9k
8
votes
Accepted
Equivalent notions of congruence for elliptic curves over $\mathbb{Q}$
I found myself wondering the same thing a couple of weeks ago. Even with the restriction that $p > 2$ and the residual representation is irreducible, it does not follow that $E_{1}$ and $E_{2}$ have t …
- 19.9k
16
votes
Accepted
$x^3+x^2y^2+y^3=7$, and solvable families of Diophantine equations
(a) No. There are no integer solutions. The curve $C$ you give has genus $3$ and it has an obvious automorphism $\phi(x,y) = (y,x)$. The quotient curve is an elliptic curve. In particular, if you let …
- 19.9k
26
votes
Accepted
Does the set of square numbers adhere to Benford's law in every base?
No. Benford's law works well for sequences that grow exponentially, and the squares grow too slowly.
In particular, fix a base $b \geq 3$, consider the case of $d = 1$, and choose $n = 2 \cdot b^{2k}$ …
- 19.9k
1
vote
Bounds on largest possible square in sum of two squares
Rather than discuss $\max b_{i}$, I'll discuss the equivalent question of bounding $\min a_{i}$. The ABC conjecture implies that for all $\epsilon > 0$, $\min a_{i} \gg (c^{2}+1)^{n/2 - 1 - \epsilon}$ …
- 19.9k
7
votes
Accepted
Can the Petersson inner product $\langle f(z), f(2z) \rangle$ be zero?
Yes, the Petersson inner product can be zero. In my paper "Explicit bounds for sums of squares (see Lemma 5) I show that if $f$ is a newform of level $N$ and $p$ is a prime that does not divide $N$, t …
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10
votes
Accepted
A strengthening of base 2 Fermat pseudoprime
Such an $n$ must be prime. If $\frac{1}{k} \binom{n-1}{2k-1}$ is an integer for all $1 \leq k \leq \lfloor \frac{n}{2} \rfloor$, then $n$ divides $\binom{n}{2k}$
for all $1 \leq k \leq \lfloor \frac{n …
- 19.9k
5
votes
What are the $j$-invariants of the genus 1 modular curves?
First, there is no comprehensive list of all models of modular curves of genus $1$. (There is a list of congruence subgroups of $SL_{2}(\mathbb{Z})$ here.) Many cases have been computed, including the …
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11
votes
Accepted
Primes dividing $2^a+2^b-1$
This is a heuristic which suggests that the problem is probably quite hard. We have that $p | 2^{a} + 2^{b} - 1$ if and only if there is some integer $k$, $1 \leq k \leq p-1$ with $k \ne \frac{p+1}{2} …
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23
votes
Accepted
Algorithmic (un-)solvability of diophantine equations of given degree with given number of v...
I'm going to take a stab at this. First (as mentioned in Andres Caicedo's answer to this question), Siegel proved in 1972 that there is an algorithm to determine whether a quadratic equation in any nu …
- 19.9k