Questions on more advanced topics of number theory. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

learn more… | top users | synonyms (1)

5
votes
3answers
37 views

How prove this $x_{1}+2x_{2}+\cdots+1990x_{1990}\neq 0$

Question: let $x_{i}=1$ or $-1$,$i=1,2,\cdots,1990$, show that $$x_{1}+2x_{2}+\cdots+1990x_{1990}\neq 0$$ this problem it seem is easy,But I think is not easy. I think note ...
0
votes
3answers
53 views

How is the set of all closed intervals countable?

I am trying to figure out the answer to the problem: Show that the set of all closed intervals $[a,b]$ with $a,b \in \mathbb{Q}$ is countable. Now I know that the interval $[0,1)$ for example is ...
0
votes
0answers
14 views

Name/properties of a difference of continuants

(This is cross-posted at http://mathoverflow.net/questions/181619/name-of-a-difference-of-continuants) Suppose that $q_1$, $\ldots$, $q_s$ is a sequence of positive integers. Denote by $[q_1, ...
0
votes
0answers
11 views

Divisibility of the forms

Any positive integer of the form $4k+3$ does not divides $y^2 + 1$ I have seen in the link of: Not a perfect square of the form for any integer x. Please explain how the above statement is true? ...
0
votes
1answer
30 views

Divisions with prime cases of equations

For an ODD integer $x$ let us define $y^2 + 1 = (x + 2) (x^2 -2x + 4)$ is there any prime $p$ such that $p\mid(x^2 -2x + 4)$ or $p\mid(x+2)$ in order to $ p|y^2 +1$
4
votes
4answers
88 views

If $r^{n-r}=k^{n-k}$, when is this true other than $r=k$?

Let $n,r,k$ be non negative integers such that $r,k\leq n$. If $r^{n-r}=k^{n-k}$, when is this true other than $r=k$ ? For example it is holds for $n=6,r=2,k=4$.
1
vote
0answers
44 views

Exercise in number theory

The problem is as follows: (a) Let $N$ be a natural number. Let $p_1,p_2,...p_k\leq N$ be all prime numbers less than or equal to N. Prove there is a unique factorization for any $n\leq N$ as ...
0
votes
1answer
33 views

divisibility of integer 7 [duplicate]

For any value of $y$, is $7|y^2 + 1?$ and $3|y^2 +1$? as well as $19|y^2 +1$? If there is no such $y$, how do you prove it. Also, I want to know about free software to check these type of ...
1
vote
1answer
365 views

question on how to decrypt the message

A message is encrypted using an affine cryptosystem in which plaintext uses the 26 letters A through Z (all blanks are omitted), the letters are identified with the residue classes of integers (mod ...
0
votes
0answers
42 views

is $x_{n}\ll \overline{x}_{n}^{2}$?

Let $(x_{n})_{n\ge 1}$ be an increasing sequence of positive integers and $\displaystyle{\overline{x}_{n}:=\dfrac{1}{n}\sum_{i=1}^{n}x_{i}}$. Suppose furthermore that $\forall\varepsilon\gt 0, \ \ ...
4
votes
4answers
65 views

Summable enumerations of $\Bbb Q$

We say that a set of natural numbers $A$ is summable if $\sum_{n\in A}\frac1n$ is finite. It is not hard to see that $\{A\subseteq\Bbb N\mid A\text{ is summable}\}$ is an ideal on $\Bbb N$: Subsets ...
3
votes
2answers
68 views

Not a perfect square of the form for any integer x.

Now a days, I become good fan of this site, as this site making me to learn more math..hahaha. Okay! Can we prove that $x^3 + 7$ cannot be perfect square for any positive/negative or odd/even ...
0
votes
0answers
11 views

limit of regular hyperbolic integrals is a unipotent integral (GL2 calculation)

In developing a simple trace formula for $G$=GL$_2$ over a number field $F$ one encounters the following identity of local integrals: $$\int_{G_v}f_v(g^{-1}\begin{pmatrix}1 & 1\\ 0 & ...
2
votes
1answer
52 views

Does there exist a finite set of polynomials which do not have roots over any prime field?

The polynomial $x^2 + 1$ has a root in $Z_p$ if and only if $p \not\equiv 3 \mod 4$, and the polynomial $x^2 + x + 1$ has a root in $Z_p$ if and only if $p \not\equiv 2 \mod 3$. So each of the ...
7
votes
2answers
109 views

Can $\sum_1^n 1/k$ be arranged so that it is an integer for infinitely many $n?$

It's well known that when $n>1:$ $$\sum_{k=1}^n \frac{1}{k}\not\in \mathbb{N}$$ But if we are allowed to rearrange the series, we can for instance can get: ...
3
votes
2answers
79 views

Prove that $\sqrt[n]{2}+\sqrt[n]{3}$ is irrational for every natural $n \ge 2$.

I want to prove that that $\sqrt[n]{2}$ + $\sqrt[n]{3}$ is irrational for every natural $n \ge 2$. I tried to use some theorem of minimal polynomials, but I get nothing. Also i tried to assume that ...
0
votes
2answers
32 views

Choosing sets yielding large sumsets

I am working on a problem which basically boils down to wanting to choose sets $A_1, \ldots, A_m$ over a group $G=(\mathbb{Z}_C,+)$, where $C$ and $m$ are constants, such that the sumsets defined with ...
1
vote
1answer
285 views

How to determine whether a number can be written as a sum of two squares?

I know the following theorems: A number can be represented as a sum of two squares precisely when $N$ is of the form $n^2 \prod p_i$ where each $p_i$ is a prime congruent to 1 mod 4 If the equation ...
3
votes
0answers
86 views

Is there any integer solutions of $3x^3+3x+7=y^3$?

$3x^3+3x+7=y^3$ $x, y \in \mathbb{N}$ Having thought about it two hours, and I'm still not sure how to show there actually aren't any integer solutions. EDIT Another formulation of this problem: ...
-4
votes
1answer
80 views

Outline approach to Collatz 3n+1 conjecture / Criticism needed

This is a sketch of an approach to proving the Collatz 3n+1 conjecture true along the following lines. Instead of trying to show there are no loops and no sequences that increase without bounds, ...
-1
votes
0answers
13 views

How to solve these standard form calculation questions? [on hold]

I need help to solve these questions and im not sure if the answer is right. 1) A microsecond is 0.000001 seconds. A computer does a calculation in 3 microseconds. How many of these calculations can ...
0
votes
0answers
20 views

Partial Summation [duplicate]

In many places, it is stated that $$\sum_{p\le x}\frac{1}{p} = \log\log x + O(1)$$ easily follows from $$\sum_{p\le x}\frac{\log p}{p} = \log x + O(1)$$ by partial summation. However, I don't see how ...
0
votes
0answers
22 views

Solving system of equations using mod math for a Hill cipher

I am having trouble eliminating these variables when I try to solve this system of equations. They may not even be the right equations, but it would be nice to see this worked out so I can try my next ...
3
votes
2answers
40 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.
-2
votes
0answers
98 views

Can $3x^3+3x+7$ be cube number?

Can $3x^3+3x+7$ be cube number when $x \in \mathbb{N}$? My conjecture is that the answer is no, but I don't know how to prove it. Can anybody help me to solve this problem?
0
votes
1answer
26 views

Proving two consecutive odd primes have at least 3 prime divisors. [on hold]

Prove that the sum of two consecutive odd primes has at least three prime divisors (not necessarily different).
31
votes
5answers
4k views

The myth of no prime formula?

Terence Tao claims: For instance, we have an exact formula for the $n^\text{th}$ square number – it is $n^2$ – but we do not have a (useful) exact formula for the $n^\text{th}$ ...
0
votes
2answers
41 views

Integer solutions of $a^b+b^a+1=3ab$

“Find all positive integer solutions $(a,b)$ for which $a^b+b^a+1=3ab$” I do not have much of a concept of how to solve these kinds of assignments. Anyway, I tried to look for divisibility ...
0
votes
1answer
30 views

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 ?
1
vote
2answers
52 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: $$\lfloor nx\rfloor = \sum_{k=0}^{n-1} {\lfloor x + \frac kn \rfloor}$$ I've already ...
-35
votes
8answers
3k views
+100

Unique Representation and The Fundamental Theorem of Arithmetic

While reading this thread Is 1 a prime number?, I recalled that The Fundamental Theorem of Arithmetic (FTA) which says that every positive integer greater than 1 can get written uniquely as a product ...
11
votes
2answers
744 views

Are primes randomly distributed?

There is a famous citation that says "It is evident that the primes are randomly distributed but, unfortunately, we don't know what 'random' means." R. C. Vaughan (February 1990) I have this very ...
-4
votes
0answers
47 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 ...
3
votes
0answers
32 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} ...
12
votes
1answer
396 views

Fibonacci, compositions, history

There are three basic families of restricted compositions (ordered partitions) that are enumerated by the Fibonacci numbers (with offsets): a) compositions with parts from the set {1,2} (e.g., 2+2 = ...
0
votes
0answers
25 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 ...
0
votes
2answers
50 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 ...
-5
votes
1answer
180 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.
1
vote
1answer
39 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 ...
1
vote
0answers
50 views

Count ways to reach Nth row

Given a N*M grid I need to reach last row with following operations : ...
0
votes
0answers
47 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 ...
6
votes
2answers
166 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 $gcd(x,y)=1$.
3
votes
1answer
345 views

Definition of a peculiar quotient group of isometries

I just wrote a text file to sum up my ideas about the Riemann Hypothesis. This text is a draft, and I don't expect people here to say if this approach is interesting or not (but if by chance you think ...
1
vote
0answers
40 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 ...
0
votes
4answers
94 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 ...
1
vote
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
votes
2answers
64 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 ...
1
vote
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 ...
3
votes
1answer
163 views

units in discrete valuation rings

Let $B/A$ be an extension of discrete valuation rings such that the purely inseparable extension of residue fields $l/k$ has a primitive element, i.e., $l=k(y)$ for some element $y$ in $l$. I want to ...
0
votes
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 ...