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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3
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1answer
46 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 ...
1
vote
1answer
17 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
56 views

Finitely Many Prime Tuples can get a Factorial

Let $k$ and $a_1,a_2 \cdots a_k$ be fixed integers, each of them being $>1$. Show that there are only finitely many $k$-tuples of primes $(p_1,p_2, \cdots p_k)$, with the following property: there ...
0
votes
1answer
12 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 ...
0
votes
0answers
14 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. ...
1
vote
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 ...
0
votes
0answers
11 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
votes
1answer
61 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)
12
votes
2answers
637 views

A Weaker Version of the ABC Conjecture

The ABC conjecture states that there are a finite number of integer triples (a,b,c) such that $\frac {\log \left( c \right)}{\log \left( \text{rad} \left( abc \right) \right)}>1+\epsilon $, where ...
1
vote
3answers
29 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 ...
10
votes
0answers
118 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 $a >1$ ...
23
votes
4answers
560 views

Is this graph connected

Define the following graph on the vertex set ${\mathbb N}_{\geq1}\>$: Two numbers $a$, $b\in {\mathbb N}_{\geq1}$ are connected by an edge (written $a \ \mathcal{R} \ b)$ if and only if $a+b \ | ...
2
votes
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
votes
3answers
104 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 ...
2
votes
1answer
127 views
+50

How prove this $3^{\frac{5^{2^n}-1}{2^{n+2}}}\equiv (-5)^{\frac{3^{2^n}-1}{2^{n+2}}}\pmod {2^{n+4}}$

Question: show that: $$3^{\frac{5^{2^n}-1}{2^{n+2}}}\equiv (-5)^{\frac{3^{2^n}-1}{2^{n+2}}}\pmod {2^{n+4}},n\geq 1$$ My idea: since I have prove $$5^{2^n}-1\equiv 0\pmod {2^{n+2}}$$ ...
1
vote
0answers
14 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
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 ...
2
votes
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.
8
votes
6answers
4k views

The product of n consecutive integers is divisible by n factorial

How can we prove that the product of n consecutive integers is divisible by n factorial? Note: In this subsequent question and the comments here the OP has clarified that he seeks a proof that ...
0
votes
2answers
48 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 ...
0
votes
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 ...
2
votes
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 ...
1
vote
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 ...
1
vote
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) * ...
2
votes
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 ...
1
vote
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. ...
4
votes
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 ...
1
vote
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?
1
vote
0answers
27 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 ...
26
votes
1answer
496 views

Evaluating the sum $\lim_{n\to \infty}\sqrt[2]{2+\sqrt[3]{2+\sqrt[4]{2+\cdots+\sqrt[n]{2}}}}$

The following nested radical $$\lim_{n\to \infty}\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}$$ is known to converge to 2. We can consider a similar nested radical where the degree of the radicals increases: ...
0
votes
1answer
113 views

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

How can i calculate the sum $\sum_{i=1}^{n} \large{\frac{i}{i+1}}$?
1
vote
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 ...
4
votes
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 ...
11
votes
2answers
1k views

Do these series converge to the Mangoldt function?

Jeffrey Shallit formulated this recurrence for me: $\displaystyle T(n,1)=1, k>1: T(n,k) = \sum\limits_{i=1}^{k-1} T(n-i,k-1)-\sum\limits_{i=1}^{k-1} T(n-i,k)$ which is the lower triangular array ...
3
votes
0answers
111 views

Successive ratios of a sequence, is this limit true?

The natural numbers $1,2,3,4,5....$ can be calculated as the row sums of the triangle $T(n,k)$ equal to $1$ if $n \geq k$ and $0$ otherwise: $$\displaystyle T = \left(\begin{matrix} ...
3
votes
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 ...
0
votes
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 ...
1
vote
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 ...
4
votes
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 ...
2
votes
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
15
votes
3answers
372 views

Equality of sums with fractional parts of the form $\sum_{k=1}^{n}k\{\frac{mk}{n}\}$

I recently encountered the following equality ($\{\}$ denotes fractional part): $$\sum_{k=1}^{65}k\left\{\frac{8k}{65}\right\}=\sum_{k=1}^{65}k\left\{\frac{18k}{65}\right\}$$ and found it very ...
3
votes
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 ...
78
votes
29answers
24k views

Best book ever on Number Theory

Which is the single best book for Number Theory that everyone who loves Mathematics should read?
1
vote
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 ...
4
votes
2answers
103 views

Difficult generating function

Define a beautiful number to be an integer of the form $a^n$, where $a\in\{3,4,5,6\}$ and $n$ is a positive integer. Prove that each integer greater than $2$ can be expressed as the sum of pairwise ...
1
vote
3answers
72 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 ...
34
votes
6answers
3k views

$x^y = y^x$ for integers $x$ and $y$

We know that $2^4 = 4^2$ and $(-2)^{-4} = (-4)^{-2}$. Is there another pair of integers $x, y$ ($x\neq y$) which satisfies the equality $x^y = y^x$?
7
votes
3answers
197 views

If $\sum\frac1{a_n}$ is convergent, then irrational?

$\{a_n\}$ is a strictly increasing sequence of positive integers such that $$\lim_{n\to\infty}\frac{a_{n+1}}{ a_n}=1$$ If $\sum\limits_{n=1}^\infty\frac1{a_n}$ is convergent, can one conclude ...
-1
votes
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 ...
4
votes
1answer
103 views

Is it true that for infinitely many values of $n$, the sum of digits of $2^n$ is greater than for $2^{n+1}$?

Let $S(n)$ denote the digit sum of the integer $n$, using base $10$. How to prove that there exist infinitely natural numbers such that $S(2^n)>S(2^{n+1})$? Remarks (by Deven Ware): This is ...