Questions on more advanced topics of number theory. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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40 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
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0answers
28 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 ...
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1answer
28 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 ...
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1answer
14 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 ...
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0answers
18 views

A Question Related to Zsigmondy's Theorem

I am wondering if there is a way to prove the following statement, which bares some resemblance to Zsigmondy's Theorem. I am not sure if the statement is true, but it seems as though it should be. ...
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0answers
134 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) = ...
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1answer
22 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 ...
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0answers
19 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 ...
0
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1answer
67 views

Number theory proof regarding primes and the number of digits of the prime [duplicate]

How would you prove that if given a prime each of whose (decimal) digits is equal to $1$, then the number of its digits is a prime. (It is not known if there exists infinitely many such prime)
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3answers
32 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 ...
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2answers
56 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
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3answers
106 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 ...
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0answers
15 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 ...
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2answers
19 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 ...
3
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1answer
47 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
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3answers
52 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.
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2answers
31 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 ...
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2answers
49 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 ...
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1answer
37 views

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

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 ...
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1answer
16 views

Evaluate the Legendre symbols (503/773) and (501/773)

Evaluate the Legendre symbols (503/773) and (501/773) my solution (501/773 ) = (((167*3))/773 ) = (167/773) * (3/773) = (773/167)*(773/3) = (105/167) * (2/3) = (3/167) * (5/167) * (7/167) * ...
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0answers
32 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 ...
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0answers
120 views

Can a Mersenne number ever be a Carmichael number?

Can a Mersenne number ever be a Carmichael number? More specifically, can a number $m$ of the form $2^n-1$ that is composite ever pass the test: $a^{m-1} \equiv 1 \mod m$ for all intergers $a >1$ ...
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1answer
26 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. ...
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0answers
61 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
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0answers
28 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 ...
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0answers
29 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 ...
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1answer
42 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?
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1answer
113 views

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

How can i calculate the sum $\sum_{i=1}^{n} \large{\frac{i}{i+1}}$?
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1answer
46 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 ...
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2answers
405 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 ...
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0answers
25 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 ...
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0answers
24 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 ...
2
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1answer
53 views

how I could find the number of zeros at the end of $(5^n-1) !$? [on hold]

how I could find the number of zeros at the end of $(5^n-1) !$ ? I would be interest for any replies or any comments
4
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5answers
124 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
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2answers
48 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 ...
3
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3answers
124 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 ...
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0answers
36 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 ...
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3answers
73 views

Use the binomial theorem to give a formula for positive integers $x_{k}$ and $y_{k}$ such that $(3 + 2\sqrt{2})^{^{x}} = x_{k} + y_{k}\sqrt{2}$.

Use the binomial theorem to give a formula for positive integers $x_{k}$ and $y_{k}$ such that $$(3 + 2\sqrt{2})^{^{x}} = x_{k} + y_{k}\sqrt{2}.$$ Is this simply just applying the binomial ...
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0answers
64 views

Strategy verifying Riemann Hypothesis? [on hold]

The basic strategy for verifying the Riemann Hypothesis Count all of the zeros of $\zeta(t)$ for $0 < t < T$ Compute an upper bound on the number of zeros of the zeta function which lie in the ...
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0answers
33 views

Iterative function eventually reaching identity

For integers $a,b$ we define $f((a,b))=(2a,b-a)$ if $a<b$ and $f((a,b))=(a-b,2b)$ if $a\geq b$. Given a natural number $n>1$ show that there exist natural numbers $m,k$ with $m<n$ such that ...
1
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1answer
38 views

Sums involving square of Moebius function

I try to estimate the following sum: $$ \sum_{n \leq x}\mu(n)^2 f(n) $$ where $\mu(n)$ is a Moebius function and $f(n)$ is some multiplicative arithmetic function. If I understand it correctly it is ...
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2answers
36 views

Proving that any common multiplication of two numbers is a multiplication of their least common multiplication

Im trying to prove that if there are to numbers $n,m$ (natural numbers), and their smallest common multipe is $k$, so that $k = n·i$ and $k = m·j$ for some $i,j$ natural numbers, any common multiple ...
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0answers
32 views

Proof of convergence of Kaprekar's Constant

I've tried googling this one a bit but nothing seems to come up, even though its considered to be a well known fact. Why does the kaprekar process of taking a 4 digit number: L, generating L' and L'' ...
0
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2answers
33 views

product of comprime numbers and UFD

It is well-known that if a product of coprime numbers is a perfect square, so are the numbers. The proof depends on fundamental theorem of arithmetic, and this implies that in a UFD, if ab is a ...
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2answers
31 views

Flip cards to get maximum sum

Given N cards where if ith card has number x on its front side then it will have -x on back side and a single operation that can be done only once that is to flip any number of cards in consecutive ...
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1answer
41 views

Find extra work done by Bob

Alice has challenegd Bob game of N puzzle.N puzzle is played on N*N grid with each cell containing distinct numbered tile from 1 to N*N-1 Except one which is empty cell and represented as 0. Move ...
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0answers
31 views

variation of the Euler $\phi$ function?

Let $n \leq m$ be positive integers. Is there a function or expression giving the cardinality of the set $\{r \in \mathbb{Z}^+| 1 \leq r \leq m, \gcd(r,n) = 1 \}$? If $n = m$, it's just $\phi(n)$.
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3answers
42 views

If algebraic $a$ has degree $n$, so does $-a$

I feel like the best way to move forward is to use a contradiction proof. Since $a$ is algebraic, and is of degree $n$, it has a minimal polynomial of degree $n$, so we can write ...
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2answers
68 views

how to find taxicab numbers but for squares?

Natural numbers that can be written as the sum of squares in two or more ways. The first ten numbers are 50, 65, 85, 125, 130, 145, 170, 185, 200, 205. $$ n = a^2 + b^2 = c^2 + d^2\\ a^2 − c^2 = d^2 ...
6
votes
2answers
58 views

minimal polynomial given an algebraic number

I am trying to find the minimal polynomial for the algebraic number $1+\sqrt{2}+\sqrt{3}$. My original thought was just let $\alpha=1+\sqrt{2}+\sqrt{3}$. The method I use though seems very ...