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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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: ...
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1answer
177 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 ...
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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 ...
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1answer
97 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) ...
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1answer
107 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 ...
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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
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1answer
190 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 ...
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1answer
281 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: $$ ...
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1answer
58 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 $x \ge 2000$: ...
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4answers
563 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 !
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4answers
216 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 ...
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1answer
386 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.
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1answer
291 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 ...
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3answers
454 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 ...
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2answers
394 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 ...
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2answers
179 views

Could you give a application of a special function on number theory or analysis?

With the best effort i have ever taken, i couldn't find a application of a special function on number theory or analysis on the internet. By the way, why is the applications of special functions in ...
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1answer
351 views

Brauer group of a finite field is trivial

I know that all finite skew fields are field. How does it follow from this fact that the Brauer group of a finite field is trivial?
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4answers
354 views

Is there an algorithm that can tell whether the power of two rational numbers is rational?

It has been known since Pythagoras that 2^(1/2) is irrational. It is also obvious that 4^(1/2) is rational. There is also a fun proof that even the power of two irrational numbers can be rational. ...
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1answer
174 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 ...
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2answers
327 views

Partition function- without duplicates

Is there a function, equivalent to the partition function, that does not allow duplication? Or, alternatively, for any N, how many partitions would there be- disallowing any that have the same integer ...
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4answers
250 views

Help to find cubics with square discriminant

If the discriminant $b^2-4c$ of the quadratic $x^2 + bx + c$ is a square then it factors. For every discriminant $d^2$ we have can parametrize them all $(b,c) = (d + 2 h,h(d+h))$. edit I realized now ...
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3answers
1k views

In every power of 3 the tens digit is an even number

How to prove that in every power of 3, with natural exponent, the tens digit is an even number? (For example, 3 ^ 5 = 243, 4 is even.)
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2answers
305 views

Does there exist $X$, $Y \in\mathbb{Q}-\mathbb{Z}$ such that $X^2+2Y^2=1$?

Are there solutions in $\mathbb{Q}-\mathbb{Z}$ to $X^2+2y^2=1$? I'm pretty sure that there isnt but Im not sure how to show this. I don't have much experience (yet) with $p$-adics.
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1answer
370 views

Finding all solutions to a Diophantine equation involving partial sums of a given series

This is a follow-up to a question posted recently. Let $$s_n = \sum_{r=1}^{n} \frac{1}{r(r+1)},$$ where we take $s_0 = 0$. The problem I am interested in is this: For fixed $k \geq 2$, find all ...
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1answer
38 views

What is $\left[\frac{1}{2}(p-1)\right]! \;(\text{mod } p)$ for $p = 4k+1$?

Theorem #114 in Hardy and Wright says if $p = 4k+3$ then $$ \left[\frac{1}{2}(p-1)\right]! \equiv (-1)^\nu \mod p$$ where $\nu = \# \{ \text{non residues mod } p\text{ less than }p/2\}$. Is ...
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3answers
88 views

Show that $a - b \mid f(a) - f(b)$

I have seen this lemma elsewhere. Suppose $A$ is a domain, and $f \in A[X]$. Prove that $$a - b \mid f(a) - f(b)$$ I need to prove this. $$f(a) - f(b) \equiv 0 \pmod{a-b}$$ basically. Let, $a ...
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1answer
42 views

Do there exist Artificial Squares?

Denote an artificial square E as a number: $$E \in \Bbb{N}| \lnot (\exists y \in \Bbb{Z} | y^2 = E) \land (For \ each \ w \in \Bbb{Z} \ \exists a_w | a_w^2 \equiv E \ \pmod w) $$ In other words ...
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0answers
47 views

How many distinct factors of $n$ are less than $x$?

For some (squarefree) integer $n$ and some integer $x$, I would like to find an expression that gives, for all $n$ and $x$, a good upper bound on the function $$f(n, x) = \sum_{d|n, d < x} 1$$ ...
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2answers
121 views

Find the last digit of $77777^{77777}$

Find the last digit of $$77777^{77777}$$ I got a pattern going for $77777^n$ for $n=1, 2, ....$ to be: $$7, 9, 3, 1$$ for $n = 1, 2, 3, 4$ respectively. The idea is: $$77777^{77777} \pmod{10}$$ ...
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0answers
85 views

A Tale of Two Quadratic Identities (Pell-like)

Question is at the end. Let all variables be integers. For some constants $a,b,c,d$, assume we have initial solution {$m,n$} to, $$a m^2 + b m n + c n^2 = d\tag{1}$$ Identity 1: $$a x^2 + b x y + ...
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1answer
69 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 ...
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192 views

Find the limit of $\sum \frac{1}{log^n(n)}$

Working on convergence and divergence of infinite series, I recently focused my attention on the summation $$\displaystyle\sum\limits_{n=2}^{\infty} \frac{1}{log^n(n)}$$ While proving the convergence ...
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1answer
148 views

A Shorter Proof of Rosser's Theorem Without Using The Prime Number Theorem

While researching on the elementary proof of Bertrand's Postulate I came to know about a theorem of Rosser's which states that $p_n$ $>$ $n$ $\text{ln}$ $n$. I have seen Rosser's original proof and ...
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1answer
70 views

The function $f(t)=2+\sin(t)+\sin(t\sqrt2)$

The function $f$ defined on $\mathbb{R}$ by $$f(t)=2+\sin(t)+\sin(t\sqrt2)$$ can never reach $0$. Can we find some sequence $(t_n)_{n\geq0}$ such that $$\lim_{n \to \infty}f(t_n)=0 \ \ \ ?$$ Or in ...
3
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1answer
187 views

Proving the Möbius Inversion theorem.

Given $g(m)=\sum_{d\mid m}f(d)$, and the identities $$\sum_{m\mid n}a_m = \sum_{m\mid n}a_{n/m}$$ $$\sum_{m\mid n}\sum_{k\mid m} a_{k,m}=\sum_{k\mid n}\sum_{l\mid (n/k)} a_{k,kl}$$ $$\sum_{d\mid ...
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1answer
174 views

Proving $\mathbb{Z}[\sqrt{-2}]$, $\mathbb{Z}[\sqrt{-1}]$, $\mathbb{Z}[\sqrt{2}]$, and $\mathbb{Z}[\sqrt{3}]$ are euclidean.

I have this short class note from my graduate number theory: THEOREM: Assume that $\vert N(x + y \sqrt d)\vert < 1$ for any two rational numbers $x$ and $y$ with $\vert x \vert \leq 1/2$ and ...
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0answers
118 views

Ratio of maximal to minimal jump in the set of angle multiples (corrected)

(This is the corrected version of the question I asked here: Ratio of maximal to minimal jump in the set of angle multiples.) Let $S^1$ be the unit circle in the complex plain. Let $d:S^1\times ...
3
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1answer
114 views

Can the equation $ax^2+by^2=cz^2$ be solved in integers (excluding trivial solutions)?

Suppose $a,b,c\in\mathbb{N}$ and are each squarefree. Is there a general solution for this equation? I found that for this equation to be soluble in integers there are three necessary and sufficient ...
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1answer
111 views

Prove or disprove: if x and y are representable as the sum of three squares, then so is xy.

Prove or disprove: if x and y are representable as the sum of three squares, then so is xy. How to prove or disprove it.I am unable to get any idea on it.It would be of great help if any one could ...
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1answer
91 views

Fermat-quotient of “order” 3: I found $68^{112} \equiv 1 \pmod {113^3}$ - are there bigger examples known?

I'm rereading an older text on fermat-quotients (see wikipedia) from which I have now the Question for $$ b^{p-1} \equiv 1 \pmod{ p^m} \qquad \text{ with $p \in \mathbb P $, $1 \lt b \lt p$ and ...
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3answers
142 views

Doubts about a nested exponents modulo n (homework)

As part of my homework I am supposed to find the remainder of the division of $2^{{14}^{45231}}$ by $31$. Using the ideas explained in calculating nested exponents modulo n I tried the following: ...
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1answer
98 views

$\frac{x^5-y^5}{x-y}=p$,give what p ,the diophantine equation is solvable

for$$\frac{x^3-y^3}{x-y}=x^2+xy+y^2=p$$$p=6k+1$give p prime, On what conditions,the diophantine equation $$\frac{x^5-y^5}{x-y}=p$$ is solvable in integers.does it have a linear expression.for ...
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0answers
93 views

special values of zeta function and L-functions

I was reading in some lectures notes about the Riemann zeta-function which takes on special values: $$\zeta(2) = \sum \frac{1}{n^2} = \frac{\pi^2}{6}$$ In fact, we can compute even values of the ...
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4answers
214 views

Find the remainder when $2^{47}$ is divided by $47$

So I am trying to work out how to find remainders in several different way. I have a few very similar question, 1.) Find the remainder when $2^{47}$ is divided by 47 So i have a solution that says ...
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1answer
284 views

My conjecture on almost integers.

Here when I was studying almost integers , I made the following conjecture - Let $x$ be a natural number then For sufficiently large $n$ (Natural number) Let $$\Omega=(\sqrt x+\lfloor \sqrt x ...
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3answers
119 views

How to calculate number of possible voting outcomes

Problem Suppose, we have a voting with n voters and m candidates. Each voter can vote for 1 candidate. How many possible outcomes of voting can be? Solution I tried For 1 candidate there will ...
3
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1answer
86 views

What the rest of the division $5^{21}$ by $127$?

What the rest of the division $5^{21}$ by $127$?
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1answer
457 views

Number of samples to predict the next number in a pseudorandom number generator

Let: $$R_{n+1} = (mR_n + b) \bmod{a} $$ Assume we know the values of $R_1, R_2, \ldots, R_L $. What is the minimum value of $L$ (if it exists) such that we can determine $R_0, m, b$ and $a$?
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1answer
73 views

Why is the Jacobi symbol in this setting a unique homomorphism?

If $D \equiv 0,1$ mod $4$ a nonzero integer, why is the map given by $\chi: (\mathbb Z / D\mathbb Z)^* \rightarrow \{-1,+1\}, \chi([p]) = (D/p)$ for odd primes $p$ not dividing $D$, a unique ...
3
votes
3answers
123 views

Raising a rational integer to a $p$-adic power

Let $p$ be a prime number and $d$ a positive integer. If $n\in \mathbb{Z}_p$ (the ring of $p$-adic integers) how would one define $d^n$? Under what conditions would this be an element of ...