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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2
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1answer
66 views

Computing $\text{Gal}(\mathbb{Q}^{ab}/\mathbb{Q})$

Algebraic class field theory tells us that $\text{Gal}(\mathbb{Q}^{ab}/\mathbb{Q})$ is isomorphic to the group of connected components of the quotient $\mathbb{Q}^{\times}\backslash ...
0
votes
3answers
288 views

1729, and related questions

I just read this paragraph: (written by G. H. Hardy, on Ramanujan) I remember once going to see him when he was lying ill at Putney. I had ridden in taxi cab number 1729 and remarked that the ...
2
votes
1answer
88 views

Finding an $n$ such that $n^2 \equiv -1 \mod p$

What is an efficient algorithm to find the first number $n$ such that $n^2 \equiv -1 \mod p$ for a prime $p$, if such an $n$ exists? Is there anything better than the brute-force approach up to $p-1 ...
3
votes
2answers
95 views

Nice polynomial reducibility: $x^n+4$

Problem: Find all $n\in \mathbb{N}$ such that $f(x)=x^n+4$ is reducible in $\mathbb{Z}[x]$. It seems $n=4k$ is the only one (the factorization follows easily from Sophie Germain's identity in this ...
7
votes
1answer
61 views

Torsion of finitely generated $\mathbb Z_ \ell$-module finite?

Let $M$ be a finitely generated $\mathbb Z_ \ell$-module, where $\ell$ is a prime number and $\mathbb Z_\ell$ is the ring of $\ell$-adic integers. Let $T$ be its torsion submodule. Is $T$ finite? In ...
3
votes
1answer
70 views

Calculate 'interference' of number patterns

I have 2 numerical series like this: $$ 144 + 25 + 27 + 29 + 31 + \cdots $$ $$ 133 + 3 + 5 + 7 +9 +11+13+\cdots $$ Is there a efficient way to find the common sum of these patterns? solution for ...
2
votes
1answer
175 views

Unique Products on a Times Table

I was looking at a 10x10 multiplication table, and I decided to count the unique products. There are 42 out of a possible 100 numbers represented. I had to wonder, why 42? I counted the 58 non-listed ...
4
votes
1answer
168 views

Squeezing $\pi(x)$ out of $\psi(x)$

Can $\pi(x)$ be written in terms of $\psi(x)$? I can only seem to approximate it: $$ ...
1
vote
2answers
61 views

Can't understand source of constant for prime counting function:

Consider the prime counting function $$ \pi(x) = \ the \ number \ of \ primes \ less \ than \ or \ equal \ to \ x$$ It is well known due to the sieve eratosthenes that given an integer $n$ and the ...
1
vote
4answers
235 views

Why didn't Fermat provide proofs of his theorems?

Apparently Fermat stated but didn't provide proofs of various theorems named after him, including Fermat's little theorem, Fermat's theorem on sums of two squares, Fertmat's polygonal number theorem, ...
1
vote
1answer
80 views

Is $\{1,1,2,3,4,5,\cdots,i,\cdots \} $ the simple continued fraction algebraic or transcendental?

Is $$1+\cfrac{1}{1+\cfrac{1}{2+\cdots}} $$ or$\{1,1,2,3,4,5,\cdots,i,\cdots \} , i\in \mathbb{N}$ the simple continued fraction algebraic or transcendental? Any reference is appreciated EDIT and ...
2
votes
2answers
111 views

Diophantine equation: $n^p+3^p=k^2$

Find all solutions to the Diophantine equation $n^p+3^p=k^2$, where $p\in \mathbb{P}$ and $n,k$ positive integers. I have tried everything, from mods to bounding to LTE; nothing seems to work on ...
3
votes
1answer
51 views

Having trouble with binary quadratic forms.

A quadratic form represents an integer $n$ if there exist $x,y\in \mathbb{Z}$ such that $f(x,y)=n$. It is proper if $\gcd{(x,y)}=1$. It is said that if $f(x,y)=n$ and $\gcd{(x,y)}=g$, then $g^2|n$. ...
0
votes
1answer
29 views

Any upper bound for $a_i$ in $\gamma =\{a_0,a_1,\dots,a_i,\dots\}$ the simple continued fraction expansion of real positive algebraic numbers?

Are there any upper bound for $a_i$ in $\gamma =\{a_0,a_1,\dots,a_i,\dots\}$ the simple continued fraction expansion of real positive algebraic numbers?
0
votes
0answers
55 views

Infinitely many prime numbers 6n-1 [duplicate]

Prove that there are infinitely many prime numbers of the form $6n-1$. I proved that there are infinitely many prime numbers but I couldn't bring it in the form given in the question. While proving ...
2
votes
0answers
48 views

What's the proof for the #integers less than $n$ that can be expressed as the sum of two squares is $\frac n{\sqrt{\log n}}$?

This result is used in the Erdos' Distance problem, in the Landau-Ramanujan constant, but I can't find a proof anywhere. http://en.wikipedia.org/wiki/Erd%C5%91s_distinct_distances_problem ...
2
votes
1answer
116 views

Algebraic Integers in $\mathbb{Q}(\sqrt{m})$ and Norms on them

I'm having a problem with a section of Niven's book the Theory Of Numbers. I am trying to show: If an integer $\alpha \in \mathbb{Q}(\sqrt{m})$ is neither zero nor a unit, prove that ...
0
votes
1answer
32 views

Expanding Base 2B representation of an integer

Consider an integer$L$ written in Base 2B which digits $$a_n a_{n-1} a_{n-2} ... a_1 B$$ Where $a_i$ are arbitrary constants such that $9 \le a_i < 2B$. I am attempting to prove that the square ...
2
votes
0answers
56 views

A Question Related to Zsigmondy's Theorem

I am wondering if there is a way to prove the following statement, which bears some resemblance to Zsigmondy's Theorem. I am not sure if the statement is true, but it seems as though it should be. ...
-1
votes
2answers
630 views

Is this a solution for the problem: $\ a^3 + b^3 = c^3\ $ has no nonzero integer solutions?

Is this a solution for the problem: $\ a^3 + b^3 = c^3\ $ has no nonzero integer solutions? Suppose $\ a^3 + b^3 = c^3,\ a,b,c \in \mathbb Z^*,\ $then: $c^3 - b ^ 3 = (c - b)((c - b) ^ 2 + 3cb) = ...
3
votes
1answer
65 views

Non unique factorization domains with prime factorizations with differing number of primes

As is well-known, $Z[\sqrt{-5}]$ is not a ufd because $6$ has more than one prime factorization in this ring: $6=2\cdot 3$ and $6=(1+\sqrt{-5})(1-\sqrt{-5})$. But both of these prime factorizations ...
2
votes
0answers
87 views

Are there any primes that are never a factor of a Carmichael number?

Is there a prime number $p$ that $p > 2$, and in which $p$ is a never a factor of any Carmichael number $C_n$: (p ∤ $C_n$) Extended this to all numbers $m$, instead of just $p$, will prove the ...
1
vote
3answers
68 views

Ideals of the residual classes $\mathbb Z_n$

Let $n$ be a positive integer and considere the ring $\mathbb Z_n$ of the residual classes modulo $n$. My intuition tells me that there is no two distinct ideals of $\mathbb Z_n$ with the same number ...
0
votes
2answers
92 views

Is 7 prime or irreducible or something else in $\mathbb{Z}_{21}$

I thought I understood prime numbers pretty well, but now I'm told about this thing called irreducible, that sometimes numbers are irreducible but not prime (like 3 in $\mathbb{Z}[\sqrt{-5}]$) and ...
3
votes
3answers
147 views

Does $x^2+x+1 \equiv 0 \pmod {997}$ have solutions? Why or why not?

I'm have difficulty solving this problem in my textbook. Does $x^2+x+1 \equiv 0\pmod{997}$ have solutions? Why or why not? I guess the first step would be $$ \begin{array}{l} (2x+1)^2 \equiv ...
1
vote
0answers
39 views

Asymptotic behavior of sums of consecutive powers (bivariate)

Are there some (bivariate) closed form formulas for the asymptotic behaviour of the sum: $$\sum_{k=1}^{n} k^d,$$ where $n$ and $d$ are large integers? I am especially interested in a lower bound of ...
0
votes
2answers
27 views

Scan through all integers within a range by incrementing constant amount

Given a natural number $k$, what is a way to find out all natural number $i$ such that, when we start with $n = 0$ and keep adding $i$ to $n$, the value $n \mod k$ traverses through all numbers ...
1
vote
1answer
152 views

Equivalent definitions of a lattice in a real vector space of finite dimension

I'm currently trying to work my way through chapter seven of Serre's book "A Course in Arithmetic" with a view to learning about modular forms. During the course of this chapter the book begins to ...
3
votes
1answer
119 views

An elementary question regarding a multiplicative character over finite fields

Reading Chapter 2 of Koblitz's Introduction to Elliptic Curves and Modular Forms, I got stuck on the following question. I would like to proceed my reading, so I would appreciate any hint to this. I ...
2
votes
3answers
63 views

Diophantine Equatiοn $x^3=2^y+15$

I would like some help with the diophantine equation $x^3=2^y+15$ I have tried working with last digits and modular arithmetic but that hasn't got me anywhere.
0
votes
2answers
88 views

Let $p$ be an odd prime and $(a,p)=1$. Show that $x^2≡a$ (mod $p)$ has solutions, then $x^2≡a$ (mod $p^n)$ always has solution , for any $n>1$.

Let $p$ be an odd prime and $(a,p)=1$. Show that $x^2≡a$ (mod $p)$ has solutions, then $x^2≡a$ (mod $p^n)$ always has solution , for any $n>1$. I have solved the first part but second part need ...
1
vote
2answers
129 views

Let $p > 3$ be a prime number. Show that $x^2 \equiv −3\mod p$ is solvable iff $p\equiv 1\mod 6$.

Let $p > 3$ be a prime number. Show that $x^2 \equiv −3\mod p$ is solvable iff $p\equiv 1\mod 6$. My try is let $a$ be a solution of $x^2 \equiv -3 \mod p$. so $a^{p-1} \equiv 1\mod p$. This ...
2
votes
1answer
108 views

Primitive roots and quadratic nonresidues modulo a prime of form $2^n+1$ [duplicate]

Let $p$ be a prime number. We call a unit $a$ in $\Bbb Z/p\Bbb Z$ a primitive root, if $\text{ord}_p(a)=p-1$. Any unit in $\Bbb Z/p\Bbb Z$ can be written as some power as some power of $a$. if $p$ is ...
2
votes
1answer
58 views

Evaluate the Legendre symbols $(\frac{503}{773})$ and $(\frac{501}{773})$

Evaluate the Legendre symbols $(\frac{503}{773})$ and $(\frac{501}{773})$. My solution: $(\frac{501}{773}) = (\frac{167 \cdot 3}{773}) = (\frac{167}{773}) \cdot (\frac{3}{773}) = (\frac{773}{167}) ...
2
votes
0answers
94 views

Bernoulli Conjecture on $B_{2^n}$

So in a recent question I was trying to prove that $2^n-1$ will never be a Carmichael number (Can a Mersenne number ever be a Carmichael number?), I was going to prove it true as long as a certain ...
30
votes
4answers
1k views

Can a Mersenne number ever be a Carmichael number?

Can a Mersenne number ever be a Carmichael number? More specifically, can a composite number $m$ of the form $2^n-1$ ever pass the test: $a^{m-1} \equiv 1 \mod m$ for all intergers $a >1$ ...
1
vote
1answer
88 views

Absolute value of a cubic Gauss sum (over Field $\mathbb{F}_p$ )

I'm interested in the quantity $|\sum_{x \in \mathbb{F}_p} \omega^{ax^3+bx^2+cx}|$ where $a \in \mathbb{F}_p^*,b,c \in \mathbb{F}_p$ and $\omega$ is a primitive $p$-th root of unity i.e. ...
4
votes
0answers
140 views

Adding Numbers Pattern

A few nights ago I couldn't sleep and so started doing this: I would take a number and add up all of its digits to get a new number and then add up all of those digits and so on until there was only ...
2
votes
0answers
52 views

Fermat pseudo primes

Is it possible for a number of the form $2^p-1$ with $p\in \mathbb{P}$ (the primes) to satisfy $3^{2^p-2}\equiv 1\pmod {2^p-1}$ and not be a prime? In other words, can a Mersenne number be a Fermat ...
1
vote
0answers
53 views

Which basis orders [for the natural numbers] have been proven?

The set $A$ of nonnegative integers is called an additive basis of order $h$ if every nonnegative integer can be written as the sum of $h$ not necessarily distinct elements of $A$. For example, the ...
1
vote
1answer
104 views

Gamma function and Gauss sums

In this Wikipedia article appears this : "Gauss sums are the analogues for finite fields of the Gamma function." What was the relation between gamma functions and non-finite fields?
0
votes
1answer
131 views

Calculating of the sum $\sum_{i=1}^{n} \large{\frac{i}{i+1}}$? [closed]

How can i calculate the sum $\sum_{i=1}^{n} \large{\frac{i}{i+1}}$?
1
vote
1answer
51 views

Validity of number theories.

Recently we talked about the Riemann hypothesis in class, and yesterday I stumbled across the Goldbach conjectures. I realized there are quite a few theories that assert a particular property to all ...
4
votes
2answers
433 views

When does -1 have a squareroot in a finite field? (-1 as a quadratic residue)

For example in $\mathbb{F}_5$, $2^2=3^2=-1$. However, in $\mathbb{F}_3$, there is no solution to $x^2=-1$. When do the squareroot(s) exist, and if they do, can we say anything about their ...
0
votes
0answers
71 views

Let f(x,y) be a positive semidefinite quadratic form with discriminant 0. Show that f is equivalent to the form h(x,y) = $gx^{^{2}}$.

Let f(x,y) = $ax^{^{2}} + bxy + cy^{^{2}}$ be a positive semidefinite quadratic form of discriminant 0. Put g = gcd(a,b,c). Show that f is equivalent to the form h(x,y) = $gx^{^{2}}$. I know that if ...
1
vote
0answers
36 views

Question about finite sums and integer recursions.

Let $n$ be a positive integer and let $g(n)$ be a given strictly increasing integer function such that $0<g(n)<n$ for all $n$. Also the sequence $ |g(n) - n|$ is unbounded as $n$ grows. Let ...
6
votes
5answers
269 views

What is the proof to the fact that all prime numbers are 1 above or below a 6 multiple? [duplicate]

I was just having an argument with my friend and I dunno how we got here. But he suddenly said all primes are 1 above or below a multiple of 6. At first I tried a lot of primes but couldn't disprove ...
3
votes
2answers
145 views

Does it hold that the $p$-adic completion of the integers equals the completion of the localization in $p$?

maybe this is a stupid question, but I could not solve it even for the ordinary integers $\mathbb{Z}$. Furthermore, I don't have to much knowledge on algebraic number theory and ramifications. Let ...
4
votes
3answers
195 views

$x^2+1$ is almost always square free

It seems like $x^2+1$ is almost always square free. Any research or heuristics why? I tried breaking the problem into solving $$x^2-ky^2=1$$ For various $k$, and I conjecture that for every $k$ there ...
3
votes
1answer
86 views

Basic Iwasawa Theory Question

I'm looking at a paper that introduces some terms and intends to use concepts from Iwasawa Theory. I instantly find myself stuck at the second sentence and even after much searching on the internet, I ...