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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Calculating a floor sum

Is there any explicit closed form expression for $\sum_{k=1}^{\dfrac{p-1}2} \bigg\lfloor \dfrac{kq}p \bigg\rfloor-\bigg\lfloor \dfrac{k(q-1)}{(p-1)} \bigg\rfloor$ , where $p,q$ are odd primes ?
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1answer
28 views

divisibility on prime and expression

This site is amazing and got good answer. This is my last one. If $4|(p-3)$ for some prime $p$, then $p|(x^2-2x+4)$. can you justify my statement? High regards to one and all.
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0answers
32 views

Identifying Symbols [on hold]

When you see $x$ written on a piece of paper you automatically identify it. When you yourself write $x^2 + 2x = 0$ The $x$ you write in $x^2$ differs from the $x$ you write in $2x$ just by a ...
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0answers
24 views

how to find the last non-zero digit of $n$

I want to know how to find the last non-zero digit of $n$. For example $n = 100!$ my try: First i have to know how much Zeros $100!$ has so i did this: $$E_{5}10 = \sum _{1\leq k <n} ...
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2answers
45 views

Proving the general formula [nx] where [.] is the floor function.

I've been trying to solve a exercise that asks me to prove the following generalization for the floor function: $$\lfloor nx\rfloor = \sum_{k=0}^{n-1} {(x + \frac kn)}$$ I've already proven the ...
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0answers
24 views

Simple number theory question involving division

I have a simple number theory question that involves a proof. Here is the question: If p divides abc and p does not divide a and p does not divide b prove that p does not divide ab. I'm sure this ...
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1answer
97 views

Do Prime Numbers have a Structure or do they sprout out Randomly among positive Integers? [duplicate]

Since the Order of Sequence of the Prime Numbers has not been found, it seems that all famous Mathematicians have opted for the random appearance of Primes.
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2answers
46 views

Are there particular techniques to find the general formula for an arithmetic function, neither multiplicative nor additive?

I was reading about the Euler phi function and the sigma function when I began to wonder how on earth one gets to the general formula for an arithmetic function. I'm not considering trivial formulae ...
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1answer
36 views

$P(n)$ is product of all digits of $n$. Find all $n$ such that $P(n)$ = $n^2−10n−22$.

$P(n)$ is defined as product of all digits of $n$ (decimal representation). Find all $n$ such that $P(n)$ = $n^2−10n−22$. I know the answer, which I will post later on in few days, but I want ...
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0answers
43 views

Door game between alice and bob

Alice and Bob are taking a walk in the Land Of Doors which is a magical place having a series of N adjacent doors that are either open or close. After a while they get bored and decide to do ...
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0answers
35 views

Finishing conclusion of GCD proof?

I'm trying to prove that $a$ divides $bc$ if and only if $$\frac{a}{\gcd(a, b)} \mid c$$ I go in the right direction first (i.e. if $a$ divides $bc$ then $\frac{a}{\gcd(a, b)} \mid c$): We want to ...
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0answers
48 views

Count ways to reach Nth row

Given a N*M grid I need to reach last row with following operations : ...
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5answers
2k views

The myth of no prime formula?

Terence Tao claims: For instance, we have an exact formula for the $n$th square number - it is $n^2$ - but we do not have an exact formula for the $n$th prime number $p_n$! "God may not play ...
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1answer
64 views

Does Zhang's result on primes makes RSA weaker?

I read from Finnish newspaper ( http://www.uusisuomi.fi/tiede-ja-ymparisto/72212-matemaattinen-ongelma-eli-2-300-vuotta-mies-subway-tiskin-takaa-ratkaisi#.VBwhYp09F2k.facebook ) the article of Zhang's ...
3
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2answers
62 views

partitions and their generating functions and Partitions of n

A partition of an integer, n, is one way of writing n as the sum of positive integers where the order of the addends (terms being added) does not matter. p(n, k) = number of partitions of n with k ...
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1answer
24 views

Gcd of two expressions

Given $$a(n) =n^2+20$$ Find the possible values of $$\gcd( a(n), a(n+1) ).$$ I tried doing this and got that the $\gcd$ of both the numbers should divide $2n+1$, but after this I am not able to get ...
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0answers
52 views

[ANSWERED]Is $\{n, n^{2} n^{3}\}$ a group under multiplication modulo $m = n + n^{2} + n^{3}$?

My number theory has been lacking, so i decided to practice it a bit. I have gotten better in the sense that i can figure out where to begin approaching a problem, but i am having trouble seeing the ...
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1answer
33 views

If $a$ and $b$ are positive integers, then $a^{1/b}$ is either irrational or an integer. [on hold]

Please prove: If $a$ and $b$ are positive integers, then $a^{1/b}$ is either irrational or an integer.Thank you.
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1answer
39 views

Integer $k$ such that $k!$ has 99 zeros

For how many positive integers $k$ does $k!$ has 99 zeros. The question is not difficult,since if $k$ the first for $k!$ to have 99 zeros, then since $k+1,\cdots,k+4$ are not divided by 5, so the ...
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1answer
28 views

Hilberts Theorem (norm group)

The theorem says the following: The map $N$ is a group homomorphisim from the multiplicative group of $\mathbb{Q}^{x}[i]$ to the multiplicative group of $\mathbb{Q}^{x}$ and has kernel $\lbrace ...
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1answer
45 views

Approximation of a rational number

I have been asked the following problem: Every real number $x$ can be written as a sum of the form $$ \sum_{i=1}^{\infty} \frac{a_n}{n^2}, $$ where $a_1\in \mathbb{Z}$, $a_2=0,1,2,3$, and $0\leq ...
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2answers
23 views

powers of coprime numbers

if p and q are coprime integers does that mean the positive integral powers of p and q are coprime as well? E.G if p and q are coprime integers does that imply p^3 and q^3 are also coprime?
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0answers
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How prove that there are $a,b,c$ such that $a \in A, b \in B, c \in C$ and $a,b,c$ (with approriate order) is a arithmetic sequence?

Let $N=\{ 1,2,3,..., 3n \}$ with $n$ is a positive integer and $A,B,C$ are three arbitrary sets such that $A \cup B \cup C = N, A \cap B = B \cap C = C \cap A = \varnothing, |A| = |B| = |C| = n $. How ...
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2answers
55 views

Making 24 with given number N

Initially we have a sequence of n integers: 1, 2, ..., n. In a single step, we can pick two of them, let's denote them a and b, erase them from the sequence, and append to the sequence either a + b, ...
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0answers
34 views

Finding a point on an elliptic curve

I have an elliptic curve with the equation $ y^2 = x^3 + ax + b $ in modulo p, where p is prime. I also have a point G on that curve. How can I find another point that isn't a multiple of G? I ...
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1answer
33 views

expository articles on special values of L functions

While searching for some notes on L functions i have seen the following statement... In mathematics, the study of special values of L-functions is a sub field of number theory devoted to generalizing ...
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3answers
92 views

Sum of the digits

Let $N$ be the greatest number that will divide $1305,4665$ and $6905$, leaving the same remainder in each case. Then what is the sum of the digits in $N$?
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1answer
23 views

Proving an inequality on the sum of $\log$ of primes.

Let $S(x)=\sum_{p\leq x} \ln(p)$ where $\sum_{p\leq x}$ denotes a summation over the positive prime numbers that are $\leq x$ Prove that $\forall n \in \mathbb N, S(2n+1)-S(n+1)\leq ...
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1answer
28 views

How can we show that $\pi (x) \leq \frac{x}{2}+1$?

What is the proof that the prime counting function $\pi (x)$ is such that $$\pi (x) \leq \frac{x}{2}+1$$
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0answers
31 views

Partition function proof

I am looking for any online information regarding Hardy and Ramanujan's proof, perhaps the proof itself, that the partition function $p(n)$ is asymptotic to $$\frac{e^{K\sqrt{n}}}{4n\sqrt{3}}$$ where ...
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0answers
55 views

Mordell Diophantine: $x^2+11=y^3$

I've been trying to solve the diophantine $$x^2+11=y^3$$ recently but to no avail. I tried the "UFD trick", re-writing as $(x-i\sqrt{11})(x+i\sqrt{11})=y^3$, but it didn't give me all the solutions. I ...
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1answer
63 views

Fermat's Last Equation

Sorry this is an amateur question but I was wondering since Andrew Wiles solved Fermat's Last Theorem what effect does this have any impact on Geometry. Does this prove in a sense Higher Order right ...
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2answers
160 views

When $x^2+6xy+y^2$ a square number?

Find all natural numbers $x$ and $y$ such that $x^2+6xy+y^2$ is a square number. For example, $(x,y)=(2,3)$ or $(x,y)=(3,10)$. Obviously, we can consider $gcd(x,y)=1$.
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0answers
69 views

Using properties of $\pi$ in the primes

We all know that $\pi$ and the prime numbers are intricately related, thanks to the work of famous mathematicians like Euler and Riemann. The irrationality of $\pi$ can ultimately be used to prove the ...
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0answers
29 views

Upper bound number of distinct prime factors

I want to prove that if $\omega (n)$ is the number of distinct prime factors of $n$ for $n>2$ we have $\omega (n) \leq \frac{\ln n}{\ln \ln n} + O(\frac{\ln n}{(\ln \ln n)^2})$. How can I do this? ...
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0answers
10 views

Name/properties of a difference of continuants

Suppose that $q_1$, $\ldots$, $q_s$ is a sequence of positive integers. Denote by $[q_1, \ldots, q_s]$ the numerator (in lowest terms) of the rational number represented by the continued fraction $$ ...
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0answers
5 views

Points of a lattice inside square of side $N$

Let $\Lambda\subseteq \mathbb Z^m$ be a full-rank lattice of index $h$. I would like to know an upper bound for the quantity $H_N=|\Lambda\cap [-N,N[^m|$ where $[-N,N[^m=\{(a_1,\dots,a_m)\in \mathbb ...
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0answers
63 views

Prove that $\sqrt{n}$ is irrational [on hold]

Question: Using fundamental theorem of integers and the fact that every natural number that is not prime, prove that $\sqrt{n}$ is irrational unless $n=m^2$ for some $m\in\mathbb N$. Here is how I ...
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0answers
7 views

How to show that $f$ is Completely multiplicative function

If $f$ is a multiplicative function and $(f\cdot \mu^{-1})^{-1}= f\cdot \mu$. Prove that $f$ is a is completely multiplicative function. $\mu$ is the Möbius and $\cdot$ is the simple product.
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4answers
854 views

Can any two irrational numbers NOT of the form (m+A) and (n-A) be added to produce a rational number?

$m$ and $n$ being rational numbers, A being an irrational number. I was wondering if two irrational numbers when added always yield an irrational number. All the counter-examples I could find were of ...
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0answers
49 views

Relation between Galois theory and Fermat primes

I am curious about a possible relation between Galois theory and Fermat primes. There is a general solution to any polynomial equation of degree less than or equal to $4$. The only Fermat primes (of ...
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4answers
89 views

How can we find the smallest number $n$ such that $2^{2^n} + 1$ is not a prime.

How can we find the smallest Fermat number (i.e. in the form $2^{2^n} + 1, n \in \mathbb N$) that is not prime and show that it is indeed not a prime? Yes, when $n=5$, it is not a prime. How can we ...
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0answers
30 views

Punctured Elliptic Curve

I've come across the word "punctured elliptic curve" here and there, but none of the basic texts on the topic (Husemoller, Silverman) define or mention it. What point is removed from the curve (the ...
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1answer
47 views

Sum over divisors of sum over coprimes

Set $n \in \mathbb{N}$ , $n>1$ . Consider the function $\phi_{1}$ as $$\phi_{1}(n)= \sum _{r=1 \atop \gcd(r,n) =1}^{n} r$$ Prove that $$\sum_{d|n} d \cdot \phi_{1}\Big(\frac{n}{d}\Big) = ...
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1answer
40 views

$\sqrt {-6}$ is not prime in $\mathbb{Z}+\mathbb{Z}\sqrt {-6}$

Suppose $\sqrt{-6}|(a+b\sqrt{-6})(c+d\sqrt{-6})$. I need to show that $\sqrt{-6}$ does not divide $(a+b\sqrt{-6})$ and does not divide $(c+d\sqrt{-6})$. I thought you might arrive at some ...
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0answers
37 views

Does a generalization of the Teichmuller-character for non-prime arguments exist?

Rereading an older article on Fermat-quotients in which I'd applied some p-adic-rationale I find now, that my method for the representation of bases $b$ which allow high fermat-quotients ...
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0answers
14 views

2 players cross out numbers from the line [on hold]

There is a line with even number of numbers written on it. Two players cross out this numbers one by one from left or right. At the end they find the sum of numbers they crossed out. The winner is one ...
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1answer
40 views

Prime number theorem lemma: prove that $\psi(x)\sim\pi(x)\log(x)$

I'm trying to follow the proof in Wikipedia that the PNT is equivalent to the assertion $\psi(x)\sim x$, by proving that $\psi(x)\sim\pi(x)\log x$, which it claims is a very simple proof. One ...
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1answer
30 views

Best self study book with answers to selected questions for analytic number theory?

All: Can anyone recommend Best self study book with answers to selected questions for analytic number theory ? If a book have no answers to questions, but if you know if some professors choose the ...
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2answers
31 views

Prove that if $\gcd(a,b)=1$ then $\gcd(a^m, b)=1$

I am using the Euclidean Algorithm (EA) for proof. Let $a>b$ and by EA we have $$ \begin{align} a=q_0 b+r_1 & & & \text{where }0\leq r_{1}<b \\ b=q_1 r_1+r_2 & & & ...