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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0
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2answers
17 views

Prove that there are no integer solutions to a given equation

Prove that $ 4x = y^2 + 1 $ has no integers solutions for $(x,y)$ By rules of divisibility: $$ a \mid b \implies \frac {b}{a} = n $$ for $a,b,n \in \mathbb{Z}$ So let, $ a=4x$, $b=y^2$, and $ c ...
1
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0answers
34 views

Calling all genius: $p_1^{e_1} p_2^{e_2}…p_k^{e_k}=e_1^{p_1} e_2^{p_2}…e_k^{p_k}$

Find all positive integer $e_i$ and prime number $p_1^{e_1} p_2^{e_2}...p_k^{e_k}=e_1^{p_1} e_2^{p_2}...e_k^{p_k}$ for $k\ge 1$. Is this impossible or what? I've been trying for at least a week. I've ...
1
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2answers
41 views

Find all primes of the form $2^{2^n} + 5$ for a nonnegative integer n

I'm a little lost on how to do this problem. It looks a lot like the definition for the Fermat numbers: $F_n = 2^{2^n} + 1$, however I'm not sure how to use that in order to find all of the primes of ...
1
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0answers
17 views

Number theory formula needed

Problem: $10$ numbers modulo $77$ are given. Denote them by $a_1, a_2, ... , a_{10}$. I want to find a surjective function $f(a_1, ... ,a_{10})$ with following condition: $f(a_1, ... ,a_{10}) \neq a_i ...
1
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1answer
20 views

Basic question regarding addition in $p$-adic integers

I just learned $p$-adic integers and I am confused about something. I was wondering if someone could possibly explain me how it is done. Suppose I have $\bar{a} = 1 + 0 \cdot p + 0 \cdot p^2 + 0 ...
-1
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0answers
32 views

show that n is prime if and only if $S(n)0$ [on hold]

show that n is prime if and only if $S(n)0$ $S(n)={[\pi.\dfrac{(n-1)!+1}{n}]}$
5
votes
2answers
156 views

Prove that there are infinitely many integer solutions to a diophantine equation

Prove that there are infinitely many integer solutions to the diophantine equation: $(x-y)^7 = x^3y^3$
6
votes
1answer
104 views

Olympiad problem: Erdos-Selfridge

The following problem is a special case of Erdos-Selfridge theorem: http://projecteuclid.org/euclid.ijm/1256050816 Problem: Prove that for any positive integer $n$, the product $(n+1)(n+2)...(n+10)$ ...
2
votes
1answer
22 views

Terminology: what is the “generic character” of a ternary quadratic form?

The title says it all: What is the "generic character" of a ternary quadratic form? Motivation: I'm reading a really old paper, and the author refers to this terminology without any further ...
0
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0answers
15 views

Count ways to paint the grid

Given a rectangular grid of dimension N x M. We need to paint the grid with black or white color such that there is no rectangle of size X x Y having same color in each cell. Find the number of ways ...
1
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1answer
17 views

Does decimal fraction has hex value?/can hex be fraction?

I was wondering if a decimal fraction could be converted into a hexadecimal fraction? I have seen it many times ? but I have been also told that decimal or binary fraction has no meaning in hex. ...
0
votes
1answer
14 views

What Legendre's Conjecture Implies About the Upper Bound For the Prime Gap Above Any Natural Number

Big O of the square root of n, or big O of n to the power of one half, is what Legendre's conjecture implies about the upper bound for the prime gap above any natural number n, right? I have seen it ...
4
votes
1answer
88 views

Does this prime generating way generate all the prime numbers?

I've thought of the following algorithm to find the entire list of prime numbers: Take a prime number $p$ to your list. $1.$ Multiply all the numbers in your list and call the number you get ...
0
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0answers
26 views

find sum of all sequences possible

i define a sequence $x_1,x_2,x_3,\cdots,x_{2011}$ of positive integers. the sequence is valid if the following conditons are met. $$x_1^n+2x_2^n+3x_3^n+\cdots+2011x_{2011}^n=a^{n+1}+1$$ where $a$ is ...
-2
votes
2answers
200 views

Distribution of the RSA numbers

Let's take a random prime $p$. For the sake of the argument let's say $\log(p)\approx 1000$. Let's suppose all numbers between $p$ and $p+1000^2$ are composites. What is the approximate probability ...
1
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0answers
21 views

What do arrows on top of variables mean (not vectors) [on hold]

Let B > 1 and d be positive integers such that d|B-1. Then $\vec{k_d} = \overleftarrow{k_d} = 1$
1
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2answers
14 views

Proving that ${\textrm{gcd}} (n, p-1) = 1$ if $p$ is the prime is the smallest prime divisor of $ n $

Show that if $n>1$ is a positive integer and $p$ is the prime is the smallest prime divisor of $ n $ then $ {\textrm{mcd}} (n, p-1) = 1$.
4
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2answers
95 views

Problems in elementary number theory and physics (and other branches)

Yesterday I started wondering if there are problems from number theory (elementary number theory in particular, but also advanced topics) which can be intuitively solved (or at least approached) by ...
3
votes
2answers
32 views

Minimum sum of set whose average of subsets is positive integer

A finite set of positive integers $A$ is called meanly if for each of its nonempty subsets the arithmetic mean of its elements is also a positive integer. In other words, $A$ is meanly if ...
0
votes
4answers
50 views

Show that some group is isomorphic to $\mathbb{Z_n}$

If $G$ has order $4$ and has an element of order $4$, then $G$ is isomorphic to $\mathbb{Z_4}$. Can someone briefly explain why this is true? I understand that $|G| = 4$, but I don't understand ...
1
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0answers
24 views

Set whose average of subsets is always square (cube, etc.)

Fix $n>1$. Is there a set $A$ consisting of $n$ (distinct) positive integers such that the average of any subset of $A$ is a square? (Feel free to replace "square" with "cube", "fourth power", ...
0
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0answers
21 views

references for number ring theory [on hold]

I am currently studying commutative algebra and in most ressources I have found, I am quite unhappy with the part devoted to the study of "standard" examples, and I find difficult to get surveys that ...
0
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0answers
31 views

Finding the Discriminant of a Root of a Cubic Polynomial [duplicate]

Hello Mathematics Community, I was hoping someone could assist me in finding the discriminant of a cubic polynomial with the following assumptions: Let $f(x)=x^3+ax+b$ be irreducible over ...
0
votes
1answer
30 views

A counting problem on the integer lattice

Let $K$ be a subset of the integer lattice $\mathbb Z^2$such that it contains elements of the form $k=(k_1,k_2) $ where $k_1,k_2$ are integers and $k_2\neq 0$. Find $m$, an integer if possible, such ...
0
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0answers
37 views

Decimal binary sequences that cannot be greater than $1$

Consider the family of sequences of the form $.012\ldots n$ for any natural number $n$. So, the sequences in this family are: $.01, .012, .0123, .01234,$ etc. Now consider to manipulate each ...
0
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0answers
26 views

Pillai equation solvability

I would like to learn an elementary method of solving Pillai equation. The equation $a^x-b^y = c$ has at most two solutions for $(x, y)$ in $\mathbb{Z} $, where $a$ and $b$ are greater than or equal ...
1
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2answers
34 views

Is $a^{p^n-1}=1\mod p$ where $p$ is prime number and $1<a<p-1$?

Is $a^{p^n-1}=1\mod p$ where $p$ is prime number and $1<a<p-1$? When $n=1$ by little fermats theorem theorem it is true. But i can't justify generaly whether it is correct or not. But when i ...
1
vote
3answers
53 views

How to find the last non-zero digit of $50!$

A week ago i made a similar question but nobody help me, i´ve been trying but i still don't get it. I want to know how to find the last non-zero digit of $50!$. my try: First i have to know how ...
0
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0answers
28 views

Induction in other algebraic structures

While reading this MSE question on real induction, as well as other articles a Google search brought me to, I suddenly became interested in this topic. Induction on the monoid of natural numbers ...
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0answers
24 views

How to decrypt a cipher with encryption rule ci=pi+n+im [on hold]

Letting p0 be the first plaintext letter, p1 the second, and so on. n and m are integers mod 26 and the i refers to the subtext on the ci and pi, so for instance the first letter in the ciphertext ...
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2answers
38 views

Is this a property of capital pi product?

$ {\displaystyle \prod_{i=1}^{N} (x_i + y_i) = \prod_{i=1}^{N} x_i + \prod_{i=1}^{N} y_i } $ ??
3
votes
1answer
36 views

Bibinomial coefficient integer

For integers $n \ge k \ge 0$ we define the bibinomial coefficient. $\left( \binom{n}{k} \right)$ by $$ \left( \binom{n}{k} \right) = \frac{n!!}{k!!(n-k)!!} .$$ What are all pairs $(n,k)$ of integers ...
2
votes
0answers
42 views

Quadruples of integers with $20^x + 14^{2y} = (x + 2y + z)^{zt}.$

Determine all quadruples $(x,y,z,t)$ of positive integers such that $$20^x + 14^{2y} = (x + 2y + z)^{zt}.$$ We can check that $20+14^2=216=(1+2+3)^3$. But how can we check if there are other ones?
1
vote
1answer
26 views

Simplify a power series

I am studying bernouli numbers and I'm having trouble condensing a power series. In particular, I'm studying the equation $$b(x)^2=(1-x)b(x)-xb'(x)$$ where ...
3
votes
3answers
74 views

Are there infinitly many primes of the form $n!+1$?

For some numbers $n!+1$ is prime, but all such numbers are not prime. For example, $5!+1 = 11\times 11$. The question is this: Are there infinitely many primes of the form $n!+1$?
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0answers
35 views

counting the good numbers

We need to calculate Good Numbers in range from $A$ to $B$ (Both inclusive). A number $N$ is said to be a good Number if it satisfy following conditions : If we extract every $2$-digit number of $N$ ...
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0answers
55 views

Exercise books in abstract algebra and number theory

I'm studying Herstein's Topics in algebra and Hardy&Wright's An introduction to the theory of numbers, and I was wondering if there are some exercise books (that is, books with solved problems and ...
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0answers
58 views

Where Are the Primes In Relation To the Perfect Squares? How Are the Perfect Squares Arranged Along the Natural Number Line? [on hold]

The question is concerning the location of any given prime which satisfies Legendre's conjecture, or simply, any given prime. Do they not all? All primes > 3 are in the pair of arithmetic progressions ...
1
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1answer
46 views

A question concerning a lower bound of $\pi (n)$

In the number theory course, I was asked to prove the following: $$2^{\pi(n)} \sqrt{n} > n \mathrm{\, for\, all\, } n > 1 \in \mathbb{N}$$, where $\pi(n)$ denotes the number of prime number ...
-2
votes
0answers
16 views

homomorphic cryptosysytem [on hold]

A multplicative homomorphic cryptosystem has an ecryption function E that satisfies the following property :E(m1) * E(m2) =E(m1*m2) where m1.m2 are messages.prove that RSA crptosustem is ...
3
votes
2answers
24 views

Discriminant of monic cubic function and integer roots

We all know that if the discriminant of a monic quadratic is a perfect square, then both of its roots will be integers. In my research, I'm interested in monic cubics, and I was wondering whether ...
1
vote
2answers
55 views

proportion of primes in a polynomial sequence

It is conjectured (Bunyakovsky) that when $P(x)$ is a polynomial from $\mathbb{Z}[X]$, irreducible, with positive leading coefficient and so that the integers $P(n)$ , $n\gt0$ do not share a common ...
4
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0answers
51 views

The “Largeness” of Subsets of the Natural Numbers

Recently, I was thinking about the Erdős conjecture on arithmetic progressions, which says that, if, for a set $A\subseteq\mathbb{N}$, the sum $\sum_{a\in A}\frac{1}a$ diverges (i.e. "A is large"), ...
0
votes
1answer
21 views

show existence of subsequence $\{a_{i_b}\}_b^{n+1}$

Suppose $\{a_n\}_{n=1}^{m^2+1}$ is a strictly increasing sequence of $n^2+1$ positive integers, show that there exist a subsequence $\{a_{i_b}\}_b^{n+1}$ of length $n+1$ such that $a_{i_k}$ is ...
0
votes
3answers
41 views

Show that there are only trivial solutions

How can I show that the only solutions of the diophantine equation $x^2+y^2=1$ are the trivial ones: $(x,y)=(0,1), (0,-1), (1,0), (-1,0)$ ? That's what I thought: $$x \equiv 0,1 \pmod 2 \Rightarrow ...
0
votes
2answers
50 views

Why does not the perfect number formula imply there are infinitely many perfect numbers?

We know the even perfect number formula is $2^{p-1}(2^p − 1)$ and it is known that the multiplication of a even number and odd number is a even number. So why can't we say there are infinitely many ...
0
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1answer
23 views

Is there also an other way, to prove that the diophantine equation has no solution?

I am looking at the following exercise: The diophantine equation $y^2=x^3+7$ has no solution. If the equation would have a solution, let $(x_0,y_0)$,then $x_0$ is odd. ( If $x_0$ is even, $x_0=2k ...
0
votes
2answers
69 views

1/x+1/y=1/2004. How to solve this one? [on hold]

How many integer pairs x<=y are there such that 1/x+1/y=1/2004. 2004(x+y)=xy and what next? Really stuck. Pls help. Problem from an 9th grade math textbook.
2
votes
1answer
348 views

Put N mices in nearest holes

N mice are playing in the desert, when one of them notices some hawks flying in the sky. It alerts the other mice who now realize that the hawks are going to attack them very soon. They are scared and ...
1
vote
3answers
49 views

How can I simplify the polynomial $x^4+1$ into quadratic factors? [on hold]

The teacher gave us a hint that this polynomial expression can be written as the multiplication or sum of quadratic factors at the most. How can I do this?