Questions on more advanced topics of number theory, such as quadratic residues, primitive roots, prime numbers, non-linear Diophantine equations, etc. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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0answers
17 views

Number of solutions of the following congruence

How do I find the number of solutions of the congruence $z^2=d(\mod 4n)$. d is the discriminant of the of the quadratic field equation $ax^2+bxy+cy^2$. I am trying to evaluate the number of terms of ...
0
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1answer
30 views

For $n$ an even number and $p$ a prime, does $\lfloor {\frac {n}{4}} \rfloor= \frac{1}{\pi (n)-\pi (n/2)} \sum_{n/2<p<n} p-\frac {n}{2}$ hold?

I was playing around with prime numbers and I noticed that for $n$ an even number, the average of the distance between all primes between $n/2$ and $n$ and $n/2$ is equal to $\lfloor {\frac {n}{4}} ...
2
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0answers
30 views

Calculating the Galois group of a covering map

Suppose $C$ is an algebraic curve and $\phi:C\rightarrow \mathbb{P}^{1}$ is a covering map of the complex projective line ramified at $\{0,1,\infty\}$ only. Suppose $\phi':C'\rightarrow ...
4
votes
6answers
155 views

Why does the $\sum_{n=1}^x \frac {1}{n} \sim \mathrm {ln} x + \frac {1}{2}$

I was playing around with the Harmonic Series and I noticed that: $$\sum_{n=1}^x \frac {1}{n} \sim \mathrm {ln} x + \frac {1}{2}$$ I wanted to know if this is just some coincidence or if it is caused ...
2
votes
1answer
56 views

Solutions of $(2x-1)^x\equiv1\mod\ p$ [closed]

Has the equation $(2x-1)^x\equiv 1\mod{p}$, for $p=1+6qx$, where $p$, $q$ are primes, $x$ is an odd integer and $x<p$ any solutions except $x=1$? Many thanks.
13
votes
1answer
134 views

“Binomiable” numbers

Is there a nice criterion to determine whether a given natural $m$ can be written as a binomial number $\binom{n}{k}$ with $1 < k < n-1$? I've been thinking on this problem with a friend and ...
3
votes
1answer
37 views

Question regarding number congruences?

First of all, before the question, I want to clear that how does $17x \equiv 1 \pmod 4 $ imply $x \equiv 1 \pmod 4$? I did: $17x \equiv 1 \pmod 4 $ $16x \equiv 0 \pmod 4$ Subtracting both, We ...
3
votes
5answers
132 views

Why is $-i^3 = i$?

Why is the value of $-i^3$ equal to $i$? After experimenting, I got this result - $-i^3=-i^2\cdot -i=1 \cdot -i=-i$ What is the error in my proof? EDIT Here is the original proof - ...
1
vote
0answers
21 views

Holomorphicity and modularity of Jacobi forms

I am reading this paper Suerconformal algebras and mock theta functions by T. Eguchi and K. Hikami. In section 3.2, the authors define a function $$J(z;w;\tau) = ...
8
votes
2answers
79 views

Find all postive integers $n$ such $(2n+7)\mid (n!-1)$

Find all postive integers $n$ such that $$(2n+7)\mid(n!-1).$$ I have $n=1,5$, but can not find any other and can not prove whether there is any other solution or not.
0
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2answers
62 views

Is the integer $0$ a deficient number?

It is well known that the divisors of the integer $0$ are all non zero-integers numbers ,the sum of those divisors greater than $0$, then is it a deficient number ? Thank you for any help
6
votes
1answer
41 views

Median order of an element in an additive group modulo $n$

I'm trying to gain some intuition here. Suppose we have the group $\mathbb{Z}_{n}$ (with the operation being addition modulo $n$). What is the median order of an element of this group when $n$ is a ...
3
votes
0answers
36 views

Cokernel of map, function field.

Let $F$ be a function field in one variable with total constant field $k$, let $X$ be the set of all places of $F$, and let $S$ be a nonempty finite subset of $X$. We are interested in the dimension ...
1
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1answer
24 views

A congruence for the prime counting function in Wolfram.What does it actually say?

I saw today in functions.wolfram.com a congruence for the prime counting function which says $\binom {2prime(k)-1} {prime(k)-1} \pmod{prime(k)^3}=1$ (the third congruence at the bottom). What does ...
0
votes
2answers
47 views

On primality of the numbers of the form $10^{2k} - 10^{(k+1)} -1$

Has anyone seen proof that numbers of the form $10^{2k} - 10^{k+1} - 1 \space \forall k \ge 2$ are prime?
2
votes
0answers
62 views

$A_4$ extension of $\mathbb{Q}$ ramified at one prime

How can one show that an $A_4$-extension of $\mathbb{Q}$ ramified at only one prime must be totally real?
1
vote
1answer
40 views

Sum of All Multiplication of Partitions

$N$ is an integer. $$ n_1 + n_2 + n_3 + \cdots + n_k \leq N $$ $$n_i > 0$$ I want to find the sum of all possible $$ n_1 n_2 n_3 \cdots n_k$$ For eg: if $N=3$ and $k=2$, answer is $$ (1\cdot1) ...
1
vote
0answers
14 views

What is a good site or book for understanding base/radix?

I have a hard time, understanding what base is, how to convert from one to another, and why is the conversion so? And I can't find a site or book that explains all these in detail.
7
votes
1answer
81 views

Is the set $\phi(\mathbb{N})$ syndetic?

A set $A \subset \mathbb{N}$ is said to be syndetic if the gaps in $A$ are bounded. Is the set $\phi(\mathbb{N})$ syndetic? (where $\phi$ denotes de Euler totient function) I've thought quite a ...
4
votes
1answer
50 views

question on identity of sums with exponent. [duplicate]

I want to show that for $x>0$: $$\sum_{n=-\infty}^\infty e^{-n^2\pi x}= \frac{1}{\sqrt{x}}\sum_{n=-\infty}^\infty e^{-n^2\pi / x}$$ It doesn't seem that a simple change of variables will do, like ...
1
vote
1answer
57 views

Proof of direct sum of ideal class group of Neukirch book

In books Neukirch, Algebraic Number Theory. I don't understand. 1) Why there exists $a$ such that $a\equiv c \ \mod \mathfrak p $ and $a\in ca_{\mathfrak p}^{-1}a_{\mathfrak q}$ for $\mathfrak ...
9
votes
0answers
161 views

Maximum integer not in $\{ ax+by : \gcd(a,b) = 1 \land a,b \ge 0 \}$

Ryan asked about a variation of the coin problem, which was whether for any coprime natural numbers $x,y$ every sufficiently large natural number is $ax+by$ for some coprime natural numbers $a,b$. ...
3
votes
1answer
26 views

Modular arithmetic: $P \cdot Q^{-1} \mod p$

I am reading an explanation to a programming competition, where one of the step is to calculate $P \cdot Q^{-1} \mod p$, where p is a prime. I was always doing this by calculating multiplicative ...
1
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0answers
39 views

Possible maps $(\mathbb Z[i]/\mathfrak p)^\times\to\mu_4$

Let $\mathfrak p$ be a maximal ideal of $\mathbb Z[i]$ not dividing 2. Is it true that the only maps from the cyclic group $(\mathbb Z[i]/\mathfrak p)^\times$ the the fourth roots of unity are powers ...
1
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0answers
31 views

Why does the number of ways that $n$ can be summed with at least one $1$ equal the partition function for $n-1$?

For some reason I was counting the number of partitions of $n$ that have at least one $1$ as an addend. The beginning sequence for these numbers, starting with $n=1$, is $\{1, 1, 2, 3, 5, 7, 11, 15, ...
4
votes
1answer
21 views

Number of ways to choose rows with inclusion condition

I have a large collection of lists consisting of 1's and 0's, each list the same length. I call each list a row. I want to know the number of ways to select rows such that their cumulative OR results ...
3
votes
3answers
69 views

Number of primes from $n!+1$ to $n!+n$

Why aren't there any primes between $n!+1$ and $n!+n$ for all $n>1$? This question was on AHSME 1969 #23, but the question is trivial because it's multiple choice. However, I have no idea how to ...
2
votes
1answer
61 views

Intersection between the sums of the first integers, primes and non primes

Conjecture : $$\left\{\sum\limits_{\begin{array}{c}k=1\\k\in\mathbb{Z}\end{array}}^nk \ |\ n\in\Bbb Z\right\} \cap \left\lbrace ...
4
votes
4answers
147 views

Semiprime numbers which, along with their prime factors, generate many semiprimes by concatenation

There's something quite interesting about the number $1191$: this number is a semiprime ($1191= 3 \cdot 397$), the concatenation of its prime factors in any order are semiprimes ($3397$ and $3973$ ...
9
votes
1answer
162 views

Ramanujan's transformation formula connected with $r_{2}(n)$

Let $r_{2}(n)$ denote the number of ways in which a positive integer $n$ can be expressed as the sum of squares of two integers. Here the sign as well as order of summands matters. Also by convention ...
2
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0answers
52 views

Finite Ramanujan expansions over a finite field

I'm wondering if we could have an analogy in finite fields. After all, the Discrete Fourier Transform (DFT) has been generalized to finite fields as well (with essentially identical properties as in ...
-4
votes
4answers
108 views

Find the the value of [closed]

Given that $f(x)+3x f (\frac{1}{x})=2(x+1)$ Find the value of $f(101)$?
4
votes
2answers
69 views

If for almost all $p \equiv 1$ (mod a) it holds that $p \equiv 1$ (mod m), then…

Let $a,m\in \mathbb N$ Suppose that for almost all primes $p \equiv 1$ (mod a) we have that $p \equiv 1$ (mod m) Can we say something about $a$ and $m$? For example $m$ divides $a$ or vice versa? I ...
6
votes
2answers
77 views

Existence of $\sqrt{-1}$ in $5$-adics, show resulting sum is convergent.

I know that to prove the existence of a square root of $-1$ in $\mathbb{Z}_5$, I can just plug $x = -5$ and $a = 1/2$ into the Taylor expansion$$(1 + x)^a = \sum_{n=0}^\infty \binom{a}{n} ...
1
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0answers
36 views

Modular fractions: $5 \big| 3- \frac 12$

I've read a lot here about how modular fractions are valid as long as the denominator is invertible, but they always cause me trouble understing this part: From the definition of congruence: $$ a ...
0
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0answers
21 views

Pairing function output that can be summed

Is there a pairing function that can take in a set of natural numbers N with a known set length and output a single natural number ...
0
votes
1answer
52 views

Prime ideals over $p\mathbb{Z}$

It's given a finite extension $F$ of $\mathbb{Q}$ and suppose it's ring of integers is in the form $\mathbb{Z}[x]$ for some integrer $x$. For each prime $p$ in $\mathbb{Z}$ proof there are at most $p$ ...
7
votes
1answer
74 views

Is there a name for this Fibonacci Identity

Last night I was trying to solve a problem and discovered an identity relating to the Fibonacci sequence $$ \left\lvert F_{i-j}F_{i+j} - F_{i-k}F_{i+k} \right\lvert = \left\lvert F_{k - j}F_{k+j} ...
2
votes
1answer
122 views

Show that the equation $y^2 = x^3 + 3$ has infinitely many rational solutions in $x$ and $y$.

Show that the equation $y^2 = x^3 + 3$ has infinitely many rational solutions in $x$ and $y$. I'm really not sure how to go about this question. I've been using trial and error and have not got ...
0
votes
1answer
20 views

Under what condition an element in the set of quadratic residues modulo a large prime is a generator

Let $p$ be a large prime. Given that $\gamma \in QR_p$, the set of all quadratic residue modulo $p$. Since $QR_p$ is cyclic so it has a generator. Under what condition $\gamma \in QR_p$ is a generator ...
0
votes
1answer
15 views

Simple extensions of local fields

Let $L/K/\mathbb Q_p$ be finite extensions of local fields and let $v_L$ and $v_K$ be normalised discrete valuations on $L$ and $K$ respectively. My question is quite a general one: If the ...
1
vote
0answers
55 views
+50

Special Case of Composite mersenne number mod p

We want to investigate if a composite mersenne number $p|2^{qb}-1$ where $p\nmid2^q-1$ ,$q,p$ are primes, $p=1+6qb,\ qb\equiv1(mod64) $ and $b$ is an odd number. In general for $$\begin{align*} ...
22
votes
3answers
2k views

Finding $(a, b, c)$ with $ab-c$, $bc-a$, and $ca-b$ being powers of $2$

This is a 2015 IMO problem. It seems difficult to solve. Find all triples of positive integers $(a, b, c)$ such that each of the numbers $ab-c$, $bc-a$, and $ca-b$ is a power of $2$. Four such ...
1
vote
0answers
27 views

Erdős' papers on Analytic Number Theőry

My adviser has often mentioned that Paul Erdős' works on Analytic Number Theory contain a myriad of techniques that any number theorist must know. What are some of his papers in Analytic Number Theory ...
3
votes
1answer
45 views

How shall I calculate $\sum\limits_{d\nmid n}\mu(d)$

Today when I was studying Apostol's Analytical Number theory, I came to know about the formula $\sum\limits_{d|n}\mu(d)=1$ if $n=1$ and $0$ otherwise. I understood the technique and then using the ...
4
votes
2answers
46 views

Closed form expression for products

How can I find a closed form expression for products of the following form $$\prod_{k=1}^n (ak^2+bk+c)\space \text{?}$$
0
votes
3answers
67 views

Calculate the exact value of the following expression [closed]

I propose the following exercise. Calculate the exact value of$$P=\dfrac{(10^{4}+324)(22^{4}+324)(34^{4}+324)(46^{4}+324)(58^{4}+324)}{(4^{4}+324)(16^{4}+324)(28^{4}+324)(40^{4}+324)(52^{4}+324)}$$ ...
3
votes
2answers
34 views

Is it known whether there are ever infinitely many primes of the form $\prod_i p_i^{n_i} + 1$ where the $p_i$ are fixed primes but the $n_i$ can vary?

So if we fix finitely many primes $p_i$, where one $p_i$ is $2$, but let the powers $n_i$ on the $p_i$ vary, is it known whether it is ever possible to have infinitely many primes of the form $\prod_i ...
3
votes
1answer
78 views

Infinite number of primes of the form $2^x \cdot 3^y + 1$?

Are there an infinite number of primes of the form $2^x \cdot 3^y + 1$? I really have no idea where to start with this. I thought of it because it would imply an affirmative answer to this recent ...
4
votes
1answer
75 views

Does Euler's $\phi$ function have the same value in arbitrarily large subsets of $\mathbb{N}$?

As my most recent question still does not have any answers and it appears to be a difficult problem, I propose the following problem (that seems easier), but which I still could not manage to solve: ...