10
votes
5answers
2k views

Disproving the claim that the numbers 1+2+4, 1+2+4+8, 1+2+4+8+16… alternate between prime and composite

I am working through an elementary number theory book and I have come across the following problem. Show the following claims are wrong: Claim 1: The sequence 1+2+4, 1+2+4+8, 1+2+4+8+16, ...
3
votes
2answers
87 views

Fastest way to find if a given number is prime

Given a random number, what would be the quickest possible way of finding out whether it was prime? Obviously, one could just iterate through the number in order to see if it was divisible by ...
-1
votes
2answers
105 views

Why is $y^{x-1}-1$ divisible by $x$?

I wanted to know if there is a way to prove that $y^{x-1}-1$ is divisible by $x$. Where $x$ is a prime number and is not equal to $y$, and $y$ is any positive whole number besides $1$. For example, ...
1
vote
0answers
34 views

A Question Related to Zsigmondy's Theorem

I am wondering if there is a way to prove the following statement, which bears some resemblance to Zsigmondy's Theorem. I am not sure if the statement is true, but it seems as though it should be. ...
1
vote
1answer
26 views

Length of smallest repunits divisible by primes

I want to prove this statement from Wikipedia: It was found very early on that for any prime p greater than 5, the period of the decimal expansion of 1/p is equal to the length of the smallest ...
1
vote
1answer
65 views

Characterizing the primes which don't divide any Pell-Lucas number(s)

For integer $n$, let $P_n$ be a Pell number, and $Q_n$ its companion. Is there a characterization of the prime numbers $p$ which don't divide any $Q_n$? By brute-force search, I found that this ...
3
votes
1answer
76 views

Probability that two random integers have only one prime factor in common

The probability that two integers picked at random are relatively prime is known to be $1/\zeta{(2)}=6/\pi^2\approx0.607927...$. Generalizing, the probability that $n$ random integers have $\gcd=1$ is ...
0
votes
1answer
56 views

The binomial coefficients $\binom{n}{ p}$ are divisible by a prime $p$ only if $n$ is a power of $p.$

I'm looking for a "high school / undergraduate" demonstration for the: All the binomial coefficients $\binom{n}{i}=\frac{n!}{i!\cdot (n-i)!}$ for all i, $0\lt i \lt n$, are divisible by a prime $p$ ...
1
vote
3answers
47 views

If there is an $a\in\mathbb{Z}$ with $a^{n-1}\equiv 1\mod n$ but $a^{\frac{n-1}p}\not\equiv 1$ for all primes $p\mid n-1$, then $n$ is a prime

Let $n\in\mathbb{N}$ with $n\ge 3$ and $a\in\mathbb{Z}$ such that $$a^{n-1}\equiv1\text{ mod } n\;\;\;\wedge\;\;\;a^{\frac{n-1}{p}}\not\equiv1\text{ mod }n\;\;\;\forall p\in\mathbb{P}:p\mid n-1$$ ...
0
votes
1answer
80 views

Why does the number of divisors of a superior highly composite number is always a highly composite number up to 720720 ? (the only exception is 120)

I've calculated the number of divisors of every superior highly composite number up to $10^{27}$: http://oi59.tinypic.com/ndaijo.jpg The number of divisors of a superior highly composite number is ...
1
vote
1answer
168 views

Why isn't $1$ a superior highly composite number?

A superior highly composite number is a positive integer $n$ for which there is an $\epsilon>0$ such that $\dfrac{d(n)}{n^\epsilon} \geq \dfrac{d(k)}{k^\epsilon}$ for all $k>1$, where the ...
0
votes
0answers
69 views

Among the superior highly composite numbers, which are the most divisor dense numbers?

I’m searching for the most divisor dense natural numbers. Firstly we have the highly composite numbers: 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, … But ...
2
votes
2answers
119 views

Asymptotic divisor function / primorials

Let $p_n\#\equiv\prod_{k=1}^{n}p_k$ (primorial), and $\sigma(n)=\sum_{d|n}^{}d$ (divisor function). Could someone please tell me what the general asymptotic of $\dfrac{\sigma(p_n\#)}{p_n\#}$ is? It ...
2
votes
3answers
40 views

Prove that if m is prime and m|kl then either m|k or m|l.

Proofs homework question, here's what I've figured out thus far. Suppose m doesn't divide k. We need to then prove that m|l. If m doesn't divide k and m is a prime then we know m and k are co-prime ...
0
votes
0answers
27 views

How to apply the generalised divisibility rule to numbers of the form $10^k+n$

This is kind of a long question but bear with me. There's actually a question mark at the end. I'm trying to apply the generalised (decimal) divisibility rules to numbers of the form $10^k+n$ where ...
1
vote
1answer
61 views

Coprimality of $2^n + 3^n$ and $5^n + 7^n$

Prove or disprove that for all positive integers $n$, $2^n +3^n$ and $5^n + 7^n$ are always coprime. This is not a homework problem, it is merely a problem I set myself after doing some number theory ...
4
votes
1answer
110 views

Primality of the number of obtained by concatenating the n consecutive digits

Let $f_n$ be the number obtained by concatenating the first $n$ numbers (in base 10). For example $f_1 = 1, f_3 = 123$ and $f_{13} = 12345678910111213.$ Now if $n$ is even or divisible by $5$ then ...
4
votes
4answers
37 views

greatest common divisor of two primes a,b

Here is the question I am trying to prove: If $a,b$ are relatively prime and a>b prove that $\gcd(a-b, a+b) \in \{1, 2\}$. Can I begin with something like $(a-b)k + (a+b)l = d$ where $k,l$ are ...
4
votes
2answers
117 views

Primes $p$ such that $p^2$ divides $x^2 + y^2 + 1$

Call a prime $p$ awesome if there exist positive integers $x$ and $y$ such that $p^2$ divides $x^2+y^2+1$. Observation: $2$ is not awesome, because $x^2+y^2+1\not\equiv 0$ (mod $4$). But $3$ is ...
7
votes
1answer
84 views

Bezouts Identity for prime powers

I have two prime powers $2^n$ and $5^n$ for some arbitrary $n$. Their gcd is $1$ but how do I get their integer linear combination which is $1$ in terms of $n$. In other words what will be the ...
0
votes
1answer
18 views

A conjecture about the existence of a member within an interval with certain divisibility conditions - counter examples?

Conjecture The interval of the natural number line $[ap_{n}, (a+1)p_{n}]$ contains a member $e$ that is not divisible by any prime number $p_{m}$ less than or equal to $p_{n}$, if $(a+1) \leq ...
1
vote
1answer
38 views

Divisibility problem with product of two primes

Be $n=pq$ a natural number product of two different primes $p,q$. Prove, that on the set $\{1.2,2.3,...,n(n+1)\}$ there are exactly 4 numbers divisible by $n$.
0
votes
1answer
75 views

How to understand Apostol's proof of the irrationality of $\sqrt{n}$ if $n$ is not a perfect square?

Recently I am reading the textbook of Apostol, Mathematical Analysis, Second Edition. On page 7, there is a theorem 1.10: If $n$ is a positive integer with is not a perfect square, then $\sqrt{n}$ is ...
2
votes
3answers
100 views

Determine the largest 3-digit prime factor of ${2000 \choose 1000}$

Determine the largest 3-digit prime factor of ${2000 \choose 1000}$. I could not approach the problem at all. I have no idea how to try the problem. Please help.
1
vote
3answers
96 views

Find all $n\in\mathbb N$, $n>3$ such that $p(n)=2n+16$, where $p(n)$ is the product of all the prime numbers less than $n$.

Find all $n\in\mathbb N$, $n>3$ such that $p(n)=2n+16$, where $p(n)$ is the product of all the prime numbers less than $n$. E.g. $p(7)=2\cdot3\cdot5$ (and $n=7$ is a solution). Let ...
0
votes
2answers
44 views

Euclidean lemma proof [duplicate]

According to Euclidean lemma it is defined that if $p$ is prime then $$p|ab\Rightarrow p|a\lor p|b$$ How to prove by descending induction that if $$p|a^n \Rightarrow p|a $$ knowing that $a^n = a ...
0
votes
1answer
74 views

Divisibility of prime numbers

I have this exercise in my worksheet in the discrete mathematics course.I don't understand the part that deals with prime numbers in integer-divisibility. "Show that for a prime number $p$, if a ...
1
vote
2answers
148 views

“Quadly” numbers with just 4 factors

A positive integer with exactly four positive factors is called "quadly". Compute the least $n$ for which each of $n,n+1$ and $n+2$ is quadly. (ARML 2008) My method of attacking this problem started ...
2
votes
2answers
64 views

What is the highest power of a prime that divides nPr?

I know that the highest power of a prime which divides $n!$ is given by $$\left[\frac np\right]+\left[\frac n{p^2}\right]+\left[\frac n{p^3}\right]...$$ Where $[x]$ is the greatest integer function. ...
3
votes
3answers
85 views

Why does $2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 + 1$ have new divisors $59$ and $509$ all of a sudden?

I am a noob when it comes to math so please bear with me. Why $2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 + 1$ has $2$ new divisors $59$ and $509$. I mean, all of its divisors are prime factors and ...
0
votes
1answer
43 views

GCD's and Proofs

Let p and q be odd primes. Prove that gcd(p + q, p - q) = 2. I have considered EEA to multiply it out, but I'm unsure where to go from there.
4
votes
1answer
69 views

Methods for finding a relatively prime integer

Here's the problem: Given a prime $p$ and an integer $x$, find an integer $c$ such that $gcd(x+c,p\#)=1$ where $p\#$ is the primorial for $p$. It is straight forward to solve this problem using ...
1
vote
2answers
65 views

Number theory proof with modular arithematic [closed]

What is the proof for: If p is an odd prime, show that $$1^n+2^n+3^n+...+(p-1)^n \equiv 0 (\mod p)$$ if $p-1$ does not divide $n$ or $\equiv -1 (\mod p)$ if $p-1$ divides $n$.
0
votes
0answers
86 views

When is a number like “ddd…ddd”+1 (where d is a digit) a perfect square or a prime?

Inspired by Is the number $333, 333, 333, 333, 333, 333, 333, 333, 334$ a perfect square?, I wonder when numbers like these are perfect squares. Certainly, all numbers of the form $000...0001$ are ...
0
votes
2answers
91 views

How to show the existence of a number with certain divisibility conditions between two multiples?

How can we show that between two even natural numbers they're exists a natural number that isn't even? How can we show that they're exists a natural number that is odd and not divisible by 3, between ...
1
vote
1answer
166 views

The proportion of numbers not divisible by prime numbers with respect to primorial numbers.

Looking at the interval of the natural numbers $ [1, p_{n}$#$] $; $\frac{1}{2}$ of the elements of this set will be even, and $\frac{1}{2}$ will be odd. $\frac{1}{3}$ of the elements of this set will ...
2
votes
1answer
75 views

A question on primes and divisibility

The question goes as follows: Prove that for any prime $p\geq 5$, $p^2-1$ will be divisible by $12$. I think I have a solution but I just wanted to double check with you guys. My attempt: If $p$ ...
2
votes
5answers
478 views

Suppose that $5\leq q\leq p$ are both prime. Prove that $24|(p^2-q^2)$. [duplicate]

Suppose that $5\leq q\leq p$ are both prime. Prove that $24|(p^2-q^2)$. This is what I got so far. I figured that since $p,q$ are bigger than $5$, there are only odd primes for this conjecture. ...
2
votes
1answer
78 views

Number of prime divisors in an interval

I've found a problem on a c++ website and I don't know how to solve it. Given an interval [x,y] and a number K I have to find how many numbers from this interval have exactly K distinct prime ...
1
vote
2answers
96 views

Distribution of primes remainders

Naively, I would expect the natural density the number of a fixed prime $p$ with remainder $m$ to the other primes to be uniform $$ d(p,m) = \limsup\limits_{n\rightarrow\infty} \frac{a(n,m)}{n} = ...
1
vote
2answers
104 views

Show that all the integer solution of $a^n = b^m$ are given by $a = t^{m/gcd(n,m)}$ , $b = t ^ {n/gcd(n,m)}$ , for some integer $t$.

The quiz of the course of number theory is coming. But I don't really know where to start to deal with this excercise. Can anyone help me? Question: Let $n$ and $m$ be positive integers. Show that ...
2
votes
0answers
80 views

An upper bound on the least common multiple of the first $2n+1$ integers

Let $p$ be a prime number and let $a, n \in \mathbb{N}$. Then $$ p^a \mid \operatorname{lcm}(1, 2, \dots, 2n+1) \implies p^a \leq 2n + 1 \implies a \leq \dfrac{\ln(2n+1)}{\ln p}$$ and ...
0
votes
0answers
86 views

Fibonacci sequense, problem od division

How to show that $7\mid F_m\Longrightarrow 8\mid m$ and $4\mid F_m\Longrightarrow 6\mid m$, knowing that (I) Two consecutive terms in the Fibonacci sequence are relatively prime. (II) In ...
0
votes
1answer
53 views

Prove that all practical numbers not of the form $2^n$ are pseudoperfect

Prove that all practical numbers not of the form $2^n$ are pseudoperfect. practical - $n$ such that every smaller integer is expressible as a sum of distinct divisors of $n$ pseudoperfect - $n$ such ...
1
vote
1answer
107 views

Divisibility of multinomial by a prime number

What is the condition for divisibility of multinomial $ \dbinom {n}{x_1, x_2, \dots, x_k} $ by a prime $p$? Update: I tried to solve using a generalisation of Lucas Theorem by representing the $n$ ...
1
vote
4answers
161 views

$(a^{n},b^{n})=(a,b)^{n}$ and $[a^{n},b^{n}]=[a,b]^{n}$?

How to show that $$(a^{n},b^{n})=(a,b)^{n}$$ and $$[a^{n},b^{n}]=[a,b]^{n}$$ without using modular arithmetic? Seems to have very interesting applications.$$$$Try: $(a^{n},b^{n})=d\Longrightarrow ...
4
votes
2answers
101 views

Valid Alternative Proof to an Elementary Number Theory question in congruences?

So, I've recently started teaching myself elementary number theory (since it does not require any specific mathematical background and it seems like a good way to keep my brain in shape until my ...
3
votes
1answer
152 views

Divisibility by a prime number

I have been struggling with this question. It would be great if somebody can really help me out with this question: Prove that for any Prime number P > 5, there exists a K such that 1111....11 ...
7
votes
3answers
136 views

Primality of the numbers in the form of $2n^2-1$

I have a question about primality of integers in the form of $2n^2-1$. I can prove that for the certain type of n such integers are always composite. For example, if $n=7k+2$ or $n=7k+5$, the whole ...
0
votes
1answer
79 views

Prime divisibility in a prime square bandtwidth

I am seeking your support for proving (or fail) formally the following homework: Let $p_j\in\Bbb P$ a prime, then any $q\in\Bbb N$ within the interval $p_j<q<p_j^2$ is prime, if and only if ...