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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3answers
607 views

Ring homomorphisms $\mathbb{Z}\to\mathbb{C}$

Let $f\colon\mathbb{Z}\to\mathbb{C}$ be a homomorphism of rings? Prove that the kernel of $f$ can not be equal to $12\mathbb{Z}$. I also wondering if the kernel can be equal to $13\mathbb{Z}$?
4
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1answer
175 views

Infinitely many transcendental numbers over Q

My previous question was not well-framed so I will ask again: Can you explicitly produce an infinite set of real numbers which is algebraically independent over $\mathbb Q$?
1
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1answer
124 views

Producing infinite family of transcendental numbers

Weierstrass proved the result [Lindemann-Weierstrass theorem] that if $a_1, \cdots, a_n$ are reals linearly independent over the rationals, then $e^{a_1}, \cdots, e^{a_n}$ are algebraically ...
0
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1answer
339 views

Calculating number of draws in a series of team matches

May be this could be an easy problem but somehow cannot arrive to any conclusive result to this indecisiveness problem For a certain number of teams ($n$) playing a series of matches ($m$), and given ...
15
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0answers
189 views

Homogeneous forms of degree $n$ in $n$ indeterminates over $\mathbb{Z}$: which ones come from the norm of a number field?

Is there a characterization of the homogeneous forms of degree $n$ in $n$ indeterminates over $\mathbb{Z}$ which occur as the norm of some algebraic number ring with a suitable $\mathbb{Z}$-basis? ...
6
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3answers
1k views

Proving Fermat's Last Theorem for n=3 using Euler's and Lamé's approach?

Euler and Lamé are said to have proven FLT for $n=3$ that is, they are believed to have shown that $x^3 + y^3 = z^3$ has no nonzero integer solutions. According to Kleiner they approached this by ...
1
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1answer
222 views

Sequence of first differences strictly increasing?

If $ \pi (x) $ := number of primes $ \leq x $, the operation $T(x_{n+1}) = x_{n+1} - \pi(x_{n+1}) = x_n$ gives a sequence whose elements are those for which repeated application of T gives the ...
0
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1answer
262 views

Computing Hermite Normal Form using Extended Euclidean Algorithm (modulo $D$)

I am attempting to compute the Hermite Normal Form of a matrix $A$, again, only this time using the Extended Euclidean Algorithm modulo the determinant of the matrix. This was a method contributed by ...
2
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2answers
1k views

Find Units in $\mathbb{Z}[\sqrt{n}]$

Find all units in $\mathbb{Z}[\sqrt n] = \{a + b\sqrt n \mid a, b \in \mathbb Z \}$ and $n \in\mathbb N$, $n\ge 2$. First let $c +d\sqrt n$ be a unit so $$(a +b\sqrt n)(c +d\sqrt n) = 1,$$ $$ac + ...
23
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1answer
897 views

Deligne, elliptic curves and modular forms

I'm trying to understand an argument of Deligne (in Courbes elliptiques: Formulaire d'après J. Tate), but I'm not familiar enough with algebraic geometry, so I'm getting quite confused. So even in my ...
3
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2answers
226 views

Statistical observations about primes

Are there statistical observations about prime numbers showing that primes are not random? For example obviously primes are $1$ or $-1$ mod $6$, but are these remainder distributed equally? What I ...
0
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3answers
393 views

Legendre Symbol, Jacobi Symbol, and Quadratic Residues..

I am looking at this problem and I am confused on how one could easily compute this. Is it just the intersection of either non-quadratic residue or quadratic residues of the respective p and q?
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2answers
130 views

Constructive proof need to know the solutions of the equations

Observe the following equations: $2x^2 + 1 = 3^n$ has two solutions $(1, 1) ~\text{and}~ (2, 2)$ $x^2 + 1 = 2 \cdot 5^n$ has two solutions $(3, 1) ~\text{and}~ (7, 2)$ $7x^2 + 11= 2 \cdot 3^n$ has ...
3
votes
1answer
127 views

Van Der Waerden without topological dynamics?

Applying topological dynamics to prove Van Der Waerden's theorem on the existence of monochromatic arithematic progression has now become a somewhat classical example of the power of topological ...
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2answers
22k views

Finding a primitive root of a prime number

How would you find a primitive root of a prime number such as 761? How do you pick the primitive roots to test? Randomly? Thanks
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1answer
426 views

Is the clustering of prime powers merely coincidental?

$2^3$ and $3^2$ are close together; $11^2$, $5^3$, and $2^7$ (121, 125, and 128) are close together; $3^5$, $2^8$, and maybe $17^2$ (243, 256, and 289) are close together. $7^3$ is close to $19^2$ ...
25
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1answer
788 views

How many primes does Euclid's proof account for?

This is a passing curiosity, and I haven't found any duplicates, so I thought I'd share my thoughts. In the most basic (or at least the most famous) proof of the infinitude of prime numbers, due to ...
2
votes
2answers
234 views

Triplets based equation

Let $p \ge 7$ be a prime number. Find the triples $(x, y, z)$ in $\mathbb{Z}$ such as $xyz$ is not equal to zero, $\gcd (x, y, z) = 1$ and $x^p + 2y^p = z^2$. I want triplets and proof/generalization. ...
7
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2answers
316 views

What does the Haar measure on $\hat{\mathbf{Z}}$ look like?

What does the Haar measure on $\hat{\mathbf{Z}} = \prod_p \mathbf{Z}_p$ look like? Does it bear any relation to the "upper density" of a subset $S\subset\mathbf{N}$ defined by $m(S) = ...
7
votes
2answers
631 views

Is the algebraic closure of a $p$-adic field complete

Let $K$ be a finite extension of $\mathbf{Q}_p$, i.e., a $p$-adic field. (Is this standard terminology?) Why is (or why isn't) an algebraic closure $\overline{K}$ complete? Maybe this holds more ...
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2answers
895 views

Find smallest number which is divisible to $N$ and its digits sums to $N$

Someone asked this question in SO: $1\le N\le 1000$ How to find the minimal positive number, that is divisible by N, and its digit sum should be equal to N. But I wonder if we don't have ...
3
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2answers
194 views

Solutions of $x^2 + 119 = 15 \cdot 2^n$ without trial and error

I seen this equation at math.stack exchange The equation $x^2 + 119 = 15 \cdot 2^n$ has only six solutions. Those are (1,3) (11, 4), (19, 5), (29, 6), (61, 8) and other one is I don't know. This ...
2
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2answers
107 views

Problem on higher order quadratic residuosity

For a prime $p$, we know cardinality of image of $Z_p$ under the map $f(x)=x^2$ is $\frac{p}{2}$. Is there any result for general polynomial like $f(x)=x^3+x$, $f(x)=x^3+x^2$ etc. ?
4
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1answer
180 views

Show $\underbrace{{111\cdots}1}_{{\small{p-1} \ 1's}}$ is divisible by $p$

What is the shortest proof to show $\underbrace{{111\cdots}1}_{{\small{p-1} \ 1's}}$ is divisible by $p$
3
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0answers
103 views

root of a unit in a real biquadratic field

Let $p_1$ and $p_2$ two primes numbers $\equiv 1\pmod 4$. If we note by $\varepsilon_m$ the fundamental unit of real quadratic field $\mathbb{Q}(\sqrt m)$, then how can be solved in ...
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1answer
138 views

Conditional equivalence of expression to cardinality of primes on square intervals

This is an exercise to show that $$\frac{\pi((x+1)^2) - \pi(x^2)}{\pi(x- \pi (x)) } \sim 1 $$ assuming the unproven hypothesis: $\displaystyle \pi (x^2, x^2+x^{2( \theta)}) \sim \frac{x^{2( ...
1
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1answer
50 views

What do you call a finite set of maps on $\mathbb{Z}$ that are closed and compatible with operations on $\mathbb{Z}$?

Let $S$ be the set of maps and $\phi,\psi \in S$. Let $x,y \in \mathbb{Z}$. Suppose that $\phi(x) * \psi(y) = \nu(xy)$ for some $\nu \in S$. Then what would you call such a system of maps? If that ...
4
votes
1answer
139 views

Let $n$ such that $\displaystyle{2^{n-2005}} | n!$

Let $n$ such that $\displaystyle{2^{n-2005}} | n!$ Prove that this number has at most $2005$ non-zero digits when written in base $2$.
8
votes
4answers
380 views

Is there a catalogue of solved Diophantine equations?

Is there a book, website or something else aiming to catalogue all or many of the Diophantine equations that have already been solved? I have two tiny books by Sierpiński in which he gives some of ...
11
votes
1answer
833 views

Fermat numbers are coprime

So today in my final for number theory I had to prove that the Fermat numbers ($F_n=2^{2^n}+1$) are coprime. I know that the standard proof uses the following: $F_n=F_1...F_{n-1}+2$ and then the ...
4
votes
1answer
93 views

Asymptotics for almost all $x$

Theorem 2.2 in Shparlinski 2006 says: For all positive integers $n\le x$ except possibly $o(x)$ of them, the bound $$M(x)\ll\frac{x}{\log x}\exp\left((C+o(1))(\log\log\log x)^2\right)$$ holds. ...
6
votes
1answer
435 views

How to show this ideal is not principal

I have been brushing up on cubic number fields. Specifically, let $s$ be a root of the polynomial $x^3 + x^2 + 3x + 17$, and consider $K = \mathbb{Q}(s)$; we have $\mathcal{O}_K = \mathbb{Z}[s]$, and ...
6
votes
0answers
208 views

Diophantine special problem

This is my another question on Diophantine equations. Prove the following great and special problem. Let $D$ and $k$ be positive integers and $p$ be a prime number such that $gcd(D, kp) = 1$. Prove ...
1
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1answer
82 views

How many natural multisets exist with a given sum?

Given natural number $n$, how many multisets are there which sum of their elements equals $n$? There is a recursive function which can give the value in $O(n^2)$, but is there a formula for that? ...
5
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0answers
279 views

Taxicab numbers.

I think most people know these numbers. Find $x,\ y,\ z,\ w$ such that $x^3 + y^3 = z^3 + w^3$ and $x,\ y,\ z,\ w$ are not equal to each other. The first is $1729$. I'm trying to figure out if ...
0
votes
2answers
1k views

Probability of an even sum

In a set of numbers there are 5 even numbers and 4 odd numbers. If two numbers are chosen at random from the set, without replacement, what is the probability that the sum of these two numbers is ...
4
votes
1answer
501 views

Diophantine Equations $z^2+2^y=3^x $ and $ z^2+4=5^n$

How one expect the possible values of $(x, y, z) = (0, 0,0), (1, 1, 1)$ and $(3, 1, 5)$ of the equation $3^x -2^y = z^2$ without by inspection. Why $n = 1$ and $3$ are valid for $5^n - 4 = z^2$. ...
4
votes
2answers
248 views

proof of a statement about the Diophantine equation $ax^2-by^2=c^2$

The Diophantine equation of the form a$x^2$ – b$y^2$ = $c^2$ with ab is not a perfect square in Z has infinitely solutions in N, provided by a particular non-trivial solution in set of N. I have ...
5
votes
0answers
135 views

Density of products of a certain set of primes

I have an infinite set S of prime numbers with relative density 0 (that is, $\lim_ns_n/p_n=\infty$ with $S=\{s_1,s_2,\ldots\}$ and $s_1 < s_2< \cdots$). I would like to find the size (in some ...
1
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1answer
101 views

A pattern in distribution of near-primes less than $2^n$

Let $\pi_k(2^n)$ be the number of almost-primes (numbers with k factors including repetitions) less than $2^n$. I noticed that for large values of n and values of k near n, a sequence $\{\pi_k\}$ ...
4
votes
2answers
292 views

Equivalence of quadratic forms over p-adic fields.

There is a theorem that states that two quadratic forms over $\mathbb{Q}_p$ are equivalent iff they have the same rank, discriminant and the same $\epsilon$ invariant. (The last is defined as ...
1
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1answer
120 views

Between 1 and $n!$, how many elements will be there which divides $n!^2$

Between 1 and $n!$, how many elements will be there which divides $n!^2$. How to dervive to the formula or any algorithm. I tried doing different combinations but not arriving to exact solutions.
3
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2answers
175 views

Number of solutions to $a_1 + a_2 + \dots + a_k = n$ where $n > 0$ and $0 < a_1 \leq a_2 \leq \dots \leq a_k$ are integers.

I know how to find the number of solutions to the equation: $$a_1 + a_2 + \dots + a_k = n$$ where $n$ is a given positive integer and $a_1$, $a_2$, $\dots$, $a_n$ are positive integers. The number ...
3
votes
2answers
664 views

“GCD” of any two real numbers

This isn't really a GCD question, because GCD is only defined for integers. I'm interested in the the existence of a common divisor of any two non-zero real numbers. In other words can you prove or ...
2
votes
3answers
371 views

Legendre Symbol Problem

I am doing revision for my number theory exam and I am stuck on the following question. Let $x$ be an even integer. Show that every prime divisor $p$ of $x^4 + 1$ satisfies $\big(\frac{-1}{p}\big)$ = ...
8
votes
2answers
935 views

GrossOne? The arXiv blog's pick of the day.

The arXiv blog having chosen GrossOne for its daily pick of today, I read the arXiv paper concerned, and posted a comment there. The arXiv blog used to be quite high profile as these things go. Is ...
3
votes
2answers
302 views

solution of the Diophantine equation of the form $(2^n)^x + p^y = z^2 $

Can we find solutions of Diophantine equations of the form : $$(2^n)^x + p^y = z^2 $$ where $k, x, y, z$ and $n$ are positive integers. -Richard Simson
4
votes
4answers
640 views

Group theory proof of Euler's theorem ($a^{\phi(m)} \equiv 1\mbox{ }(\mbox{mod }m)$ if $\gcd(a,m)=1$)

From A Classical Introduction to Modern Number Theory by Ireland and Rosen, page 33: Corollary 1 (Euler's Theorem). If $(a,m) = 1$, then $a^{\phi(m)} \equiv 1\,(m)$. Proof. The units in ...
3
votes
2answers
336 views

Finding the unit digit of the following expressions, A and B.

Let A be:$${2013}^{{2012}^{{2011}^{2010.....}}}$$ This, goes, on, till, $3^{{2}^1}$ Let B be:$${2012}^{{2011}^{{2010}^{2009.....}}}$$ This, goes, on, till, $3^{{2}^1}$, For A , I did this, Let ...
2
votes
2answers
41 views

questions about Jacobosthal number.

Jacobosthal number is defined by $J_n=J_{n-1}+2J_{n-2},J_0=0,J_1=1$. The leading part of the sequence is $0,1,1,3,5,11,21,43,85,\dots\;$. How to show that this number is the number of numbers which is ...