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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6
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2answers
126 views

Goldbach's conjecture and number of ways in which an even number can be expressed as a sum of two primes

Is there a functon that counts the number of ways in which an even number can be expressed as a sum of two primes?
6
votes
4answers
317 views

Almost a perfect cuboid

While reading a very old book on diophantine equations, I came across this exercise: Find an infinite number of positive integer solutions of the equations $$x^2 + y^2 = u^2$$ $$y^2 + z^2 = v^2$$ ...
6
votes
2answers
564 views

Is there any famous number theory conjecture proven impossible to be find out the truth or false?

Is there any famous number theory conjecture proven undecidable? Is there any history about it? i would like to know any number theory conjecture by the types of undecidable.
6
votes
1answer
431 views

Is it known or new? [duplicate]

Possible Duplicate: Starting digits of 2^n While I was randomly working with number patterns, I came along with some interesting pattern which seems to turn to a conjecture in fact. My ...
6
votes
1answer
485 views

Pythagorean triplets

Respected Mathematicians, For Pythagorean triplets $(a,b,c)$, if $c$ is odd then any one of $a$ and $b$ is odd. Here $(a, b, c)$ is a Pythagorean triplet with $c^2 = a^2 + b^2$. Now, I will ...
6
votes
1answer
290 views

Can a Pratt certificate for a prime be found in polynomial time?

Can a Pratt certificate for a prime be found in polynomial time? I guess this is the same as asking whether the AKS primality test provides extra information that allows $p-1$ to be factored quickly. ...
6
votes
1answer
344 views

Integral closure of p-adic integers in maximal unramified extension

Let $\mathbb Q_p$ be the field of p-adic numbers, and let $\mathbb Q_p^{\text{unr}}$ be maximal unramified extension in some algebraic closure of $\mathbb Q_p$. My understanding is that $\mathbb ...
5
votes
2answers
75 views

Integral solutions to $56u^2 + 12 u + 1 = w^3.$

I would like to find all integer solutions to $$56u^2 + 12 u + 1 = w^3.$$ My computer thinks the only integral point is $(0,1).$ This problem arises from Integer solutions of $x^3 = 7y^3 + 6 y^2+2 ...
5
votes
1answer
73 views

Compute $v_2\left(2005^{2^{100}}-2003^{2^{100}}\right)$

Compute $v_2\left(2005^{2^{100}}-2003^{2^{100}}\right)$ where $v_2(n)$ is the largest power of $2$ dividing $n$. I think one way to solve this is to use the binomial theorem with $2005=2003+2$, but ...
5
votes
1answer
102 views

With $N$ a constant $>0$, show $\prod_{p<x}\frac{1}{p^{N+1}-1}>\frac{0.2}{\log^2 x}$.

Related. Show that if $x$ is large enough,$$\prod_{\substack{p<x \\ p \ \text{prime}}}\frac{1}{p^{N+1}-1}>\frac{0.2}{\log^2 x}.$$ Speaking of which, Theorem 6.12, and maybe others, of this ...
5
votes
1answer
128 views

What was Lame's proof?

In 1847, Lame gave a false proof of Fermat's Last Theorem by assuming that $\mathbb{Z}[r]$ is a UFD where $r$ is a primitive $p$th root of unity. The best description I've found is in the book ...
5
votes
2answers
151 views

Can an odd perfect number be divisible by $5313$?

I know that an odd perfect number cannot be divisible by $105$ or $825$. I wonder if that's also the case for $5313$.
5
votes
1answer
103 views

For $n\ge4$, prove that $1!+2!+\cdots+n!$ cannot be the square of a positive integer

I'm trying to prove this by induction but seem to be getting nowhere.
5
votes
1answer
548 views

Divisibility of binomial coefficient by prime power - Kummer's theorem

Let's say we have binomial coefficient $\binom{n}{m}$. And we need to find the greatest power of prime $p$ that divides it. Usually Kummer's theorem is stated in terms of the number of carries you ...
5
votes
1answer
185 views

A conjecture about the difference between consecutive primes with respect to a prime number squared.

Conjecture If we have two consecutive prime numbers $p_{a}$ and $p_{a+1}$, and two other consecutive primes $p_n$ and $p_{n+1}$, so that $p_{a} < p_{a+1} < p^2_{n+1}$, then $p_{a+1} - ...
5
votes
5answers
904 views

Show that for any positive integer n, $(3n)!/(3!)^n$ is an integer.

This is also a question on my exam paper that i proved by using mathematical induction. However, my tutor tells me that it can be proved without using mathematical induction. I really want to know how ...
5
votes
3answers
117 views

How is $x^2 + x + 1$ reducible in $\mathbb{Z}_3[x]$?

I am going through my number theory notes and have got on to the bit about the ring $\mathbb{Z}_p[x]$, where $p$ is prime, and unique factorisation domains. The example I am looking at is to do with ...
5
votes
1answer
213 views

A Gauss sum like summation

I would like to calculate the following sum. Let $\zeta$ be a primitive $n$ th root of unity for some integer $n$. Here $n$ is not necessarily prime. The sum is $$\sum_{j=1}^n (-1)^j ...
5
votes
1answer
149 views

What is the analytic continuation of a multifactorial?

The $\Gamma$ function is the analytic continuation of the factorial function. Is there a similar analog for multifactorials? I am particularly interested in the double factorial. All Google has ...
5
votes
1answer
126 views

Number of solutions to equation.

The question reads as follows: Let $p$ and $k$ be positive integers such that $p$ is prime and $k > 1$. Prove that there is at most one pair $(x, y)$ of positive integers such that $x^k + px = ...
5
votes
1answer
110 views

$3^2 \ 5^2 \ldots (p-2)^2 \equiv (-1)^{\frac{p+1}{2}} \ (\mathrm{mod} \ p)$ [duplicate]

Possible Duplicate: Why is the square of all odds less than an odd prime $p$ congruent to $(-1)^{(p+1)/(2)}\pmod p$? Why is $3^2 \ 5^2 \ldots (p-2)^2 \equiv (-1)^{\frac{p+1}{2}} \ ...
5
votes
1answer
176 views

The bound of valuation

Let $V_p$ be the $p$-adic valuation. We know that $(p - 1)! + 1\equiv0\mod p$ for the prime $p$ by Wilson's theorem. I wonder if there is an upper bound for $V_p((p - 1)! + 1)$. Also I do not know ...
5
votes
1answer
180 views

Primality Testing in $\mathcal{O}_{\mathbb{Q}(\sqrt{d})}$?

What is known about the computational complexity of primality testing in $\mathcal{O}_{\mathbb{Q}(\sqrt{d})}$ where $d$ is a square-free number? For what values of $d$ is primality testing easy ...
5
votes
2answers
274 views

Convergence of $\sum\limits_{k=1}^{\infty} \frac{1}{p_{k^2}}$, where $p_k$ is the $k$th prime

Two brief questions. This seems true but I don't find it using Google. (1) Isn't $$\sum_{k=1}^{\infty} \frac{1}{p(k^2)}$$, in which $p(k^2)$ is the $k^2$th prime, known to converge? I expected to ...
5
votes
2answers
572 views

Why does the natural ring homomorphism induce a surjective group homomorphism of units?

I'm trying an old problem here: http://www.math.dartmouth.edu/archive/m111s09/public_html/homework-posted/hw1.pdf Suppose $n\mid m$, and I have a natural ring homomorphism $\varphi\colon ...
5
votes
3answers
475 views

Powers as a complete residue system modulo $p$?

Question 1. With $0 < a < p$, $p$ prime and $\gcd(a,p-1)=1$, is it true that $0, 1, 2^a, ...,(p-1)^a$ is a complete residue system modulo $p$? If not, will a similar statement hold? Question ...
5
votes
2answers
634 views

distance between consecutive primes (related to Polignac's conjecture)

Is there an elementary(or not) proof that there are at least two consecutive primes which have difference $2n$ for every natural number $n$? i remind that Polignac's conjecture states that there ...
5
votes
1answer
240 views

Does the ring of integers have the following property?

As a follow-up to this question, I'd like to ask: What are examples of rings $R$ with the property that for all finite sets of ideals $I_1,\ldots,I_n$ in $R$ the sequence $$ \bigoplus_{1\leq j < ...
5
votes
5answers
3k views

Sum of rational numbers

The sum of a finite number of rational numbers is of course a rational number, but the sum of an infinite number of rational numbers might be an irrational number. Can someone give me some intuition ...
4
votes
1answer
91 views

Sign of Ramanujan $\tau$ function

The Ramanujan $\tau(n)$ seemed to have random positive/negative signs: ...
4
votes
1answer
76 views

Eulers totient function divided by $n$, counting numbers in the set [1,m] that are coprime to n

If we divide Euler's totient function $\omega(n)$ by $n$, we obtain a fraction. If we multiply this fraction by any natural number $m$ which gives us another natural number $p$, is it true that $p$ is ...
4
votes
3answers
268 views

Upper bound for Euler's totient function on composite numbers

I've seen before the general bound $\phi(n) \leq n - n^{1/2}$ for composite $n$. Can this bound be improved at least for those $n$ when we don't have equality above? Say could we possibly have at ...
4
votes
1answer
55 views

What is $\varlimsup \frac{\omega(n)}{\log n}$?

$\omega(n)$ is the number of distinct prime divisors of $n$. How to figure out? $$\varlimsup_{n\to\infty} \frac{\omega(n)}{\log n}$$ or $ \dfrac{\omega(n)}{\log n}$ is convergent, so ...
4
votes
2answers
233 views

Formula for prime counting function

I saw this formula on this paper page 2 $$\pi (n)=\sum_{j=2}^{n}\frac{\sin^{2}\left(\pi \frac{(j-1)!^{2}}{j}\right)}{\sin^{2}(\frac{\pi }{j})}$$ Where $\pi(n)$ is the prime counting function. Is ...
4
votes
0answers
72 views

Asymptotics for Mertens function

It seems that the cumulative mean of the Mertens function is very similar in behaviour to $x$ raised to the power of the first zeta zero. I tentatively notate it as: ...
4
votes
1answer
201 views

What is the sum of the reciprocal of primes? (Yes, it diverges) [duplicate]

It's well known that the summation over 1/p diverges just as 1/n does. However, in the case of the sum of 1/n, we can establish upper and lower bounds to the sum with the integrals over 1/n and ...
4
votes
2answers
255 views

A conjecture on $\phi(n)$

Let $\phi(n)$ denote the Euler totient function of $n $. Then let $N$ be a number such that $\phi(N)$ divides $N$ . Also let $I_1= \frac{N}{\phi(N)}$ which is defined as the "Second order Index of ...
4
votes
1answer
98 views

$m \in \{2,6,42,1806,…\} $ - a problem of sum-of-$m$'th powers modulo $m$

(continuing the work for an answer for a question here in MSE and also in MO) I'm (re-)viewing the function $$ f(m) = \sum_{k=0}^{m-1} k^m $$ considering its residue modulo $m$: $$ r(m) \equiv f(m) ...
4
votes
1answer
111 views

Bernoulli polynomial root symmetry

New @ Antonio Vargas - Many thanks - feeling a little foolish! Old Can anyone point me in the direction of anything that might explain the sudden change in near-symmetrical complex roots of ...
4
votes
1answer
85 views

Expressing Eisenstein series E_k in terms of E_4 and E_6

Given an Eisenstein series $E_k$ (of level 1), it is a polynomial $P_k(E_4,E_6)$ in $E_4$ and $E_6$, and http://en.wikipedia.org/wiki/Eisenstein_series#Recurrence_relation should give a finite ...
4
votes
1answer
193 views

Legendre Symbol - Find Prime $p$ Which Divides A Polynomial

I need to find a general form of a prime number $p$ which divides the polynomial $x^2-6$, i.e. $p$ such that $x^2 - 6\equiv 0\text{ (mod }p)$. By Legendre symbol, I actually need to find a prime p ...
4
votes
1answer
292 views

Property of natural numbers involving the sum of digits

How can you prove that every natural number $M$ or $M+1$ can be written as $k + \operatorname{Sum}(k)$, where $\operatorname{Sum}(k)$ represents the sum of the digits of some number k. Example: $$ ...
4
votes
1answer
59 views

Looking for help in understanding Jitsuro Nagura's analysis of the upper bound for $\psi(x)$

I'm working on understanding Nagura's analysis of the upper bound for $\psi(x)$ which is done in Lemma 2. I am unclear on one step of his reasoning. With Lemma 1, he establishes for $x \ge 2000$: ...
4
votes
4answers
580 views

Why 4 is not a primitive root modulo p for any prime p?

I wonder why 4 is not a primitive root for any prime p ? I've been trying to find an answer with no success so far. Any suggestion would be very helpful, thanks in advance !
4
votes
4answers
221 views

Profinite and p-adic interpolation of Fibonacci numbers

On the topic of profinite integers $\hat{\bf Z}$ and Fibonacci numbers $F_n$, Lenstra says (here & here) For each profinite integer $s$, one can in a natural way define the $s$th Fibonacci ...
4
votes
1answer
392 views

Square roots of integers and cyclotomic fields

For every $ N \in \mathbb Z$ there exists an integer $n$ such that $ \sqrt N \in \mathbb Q(\zeta_n)$. I am struggling where to start this question, please suggest me few hints.
4
votes
1answer
299 views

Von mangoldt function dirichlet series

The dirichlet series for the Vonmangoldt function, $\Lambda(n)$, which is equal to zero when $n$ is not a prime a power, and $ln(p)$ when it is a prime power say, $n=p^j$, is ...
4
votes
3answers
201 views

Show that there exists infinitely many numbers that are coprime pairwise, in the set defined as following [duplicate]

The set $A$ = {$X_n\mid n\in \Bbb N$} where $X_n = a^{n+1} + a^{n} - 1$, with $a \gt 1, a \in \Bbb Z$. Show that there are infinitely many numbers that are pairwise coprime.
4
votes
3answers
466 views

Characterising reals with terminating decimal expansions

Show that a number has a terminating decimal expansion if and only if, it is rational and when in lowest terms, its denominator is coprime to all primes other than $2$ and $5$. This is an unsolved ...
4
votes
2answers
406 views

Square root modulo $p$

I've been working on this problem for a while, but hit a dead end. Here's the problem: Suppose $p$ is an odd prime. Also let $b^2 \equiv a \pmod p$ and $p$ does not divide $a$. Prove there exists ...