1
vote
5answers
58 views

prove that $3$ does not divide $n^2+1$

How do I prove that $3$ does not divide $n^2+1$, for all $n\in\mathbb{Z}$, thought of in separate cases, but did not get, induction also was unable to ....
1
vote
0answers
17 views

Almost perfect numbers

A positive integer $n$ is called almost perfect if the sum of its divisors smaller than $n$ is $n-1$. What are all almost perfect numbers $n$ such that some power $n^k$ is also almost perfect for at ...
3
votes
1answer
41 views

Squares modulo 2^n

How many squares are there modulo $2^n$? If we would deal with $p^n$, where p an odd prime, then we could use Hensel's Lemma, which clearly doesn't work with $2^n$.
1
vote
3answers
61 views

Prove that $7^{100}+3^{10}=8^{100}$ or $7^{100}+3^{10}<8^{100}$ [on hold]

Prove that $7^{100}+3^{10}=8^{100}$ or $7^{100}+3^{10}<8^{100}$ I tried using some theorems of divisibility, to show that one divides the other, and the other also divides the first, but could ...
0
votes
0answers
31 views

Find all the primes $p$ for which $x^2\equiv13\pmod p$ has a solution.

I found that for $p=3$, we have $x^2\equiv2^2\equiv4\equiv13\pmod 3\equiv-9\equiv0\pmod 3$. But how do I find out all the primes such that this holds?
2
votes
0answers
19 views

Question on Fermat Numbers Factorization

Let $F_{n}=2^{2^n}+1$ be a Fermat number. A classic idea using orders and Fermat's Little Theorem shows that a prime divisor $p$ of $F_{n}$ must be of the form $p=k .2^{n+1}+1$. Furthermore, using the ...
2
votes
2answers
79 views

for what integers $x$ do there exist $x$ consecutives integers, the sum of whose squares is prime?

for what integers $x$ do there exist $x$ consecutives integers, the sum of whose squares is prime? I tried use $$1^2+2^2+...+n^2=\frac {n(n+1)(2n+1)}{6}$$
2
votes
2answers
34 views

Determine if $-42$ is a quadratic residue of $mod(61)$

This is what I have so far: Using Legendre symbol, we have $(\frac{-42}{61})\equiv(\frac{19}{61}).$ Since $gcd(19,61)=1,$ $(\frac{19}{61})\equiv1.$ Is this correct?
1
vote
0answers
22 views

When can a congruence relation be transformed into quadratic reciprocity expressions?

When can a congruence relation $$p \equiv c_1, c_2, \ldots, c_r \mod{N}$$ be transformed back into quadratic reciprocity expressions $$\left (\frac{d_1}{p} \right) = \left (\frac{d_1}{p} \right) = ...
0
votes
1answer
34 views

Find the natural numbers so that n=2*a^2 ,n=3*b^3 ,n=5*c^5.Number theory problem.

Well here it is i spend almost 3 hours on this one!! Find the general form of the natural numbers that are twice a square ,tripple of a cube and 5 times a 5-ith power.Who is the smaller of them?.What ...
3
votes
1answer
60 views

Positive integer solutions to $x^2+y^2+x+y+1=xyz$

The question asks for positive integer solutions to $x^2+y^2+x+y+1=xyz$ . We at first note that $x|y^2+y+1$. Now,let there exist positive integers $x,y$ that satisfy the given equation.Then ...
1
vote
1answer
36 views

Rational vs irrational

If two points on a number line is shown, are rational numbers between the two points is more or irrational number is more ? I have tried using probability , my collegue who was like my teacher also ...
2
votes
1answer
49 views

Finding all primes $(p,q)$ for perfect squares.

Find all prime pairs $(p,q)$ such that $2p-1, 2q-1, 2pq-1$ are all perfect squares. Source: St.Petersburg Olympiad 2011 I have only found the pair $(5,5)$ so I am thinking that maybe a modulo $5$ ...
0
votes
1answer
22 views

Number Theory Prime Factor Problem

There is an integer N that has 12 factors, including 1 and itself, but only 3 of them are prime factors. The sum of these three prime factors is 20. What is the smallest possible value for N?
1
vote
2answers
63 views

Last Two Digits Problem

I'm trying to find the last two digits of ${2012}^{2012}$. I know you can use (mod 100) to find them, but I'm not quite sure how to apply this. Can someone please explain it?
0
votes
3answers
33 views

Number Theory Remainder Question

I'm trying to find the answer to the following: What is the remainder when 9^2012 is divided by 11? Apparently, you're supposed to use Fermat's Little Theorem, but I'm not sure how to use it to solve ...
0
votes
1answer
19 views

Equation with a sum for the prime-counting function involving the Mobius function

I have come across the statement that $$ \sum_{n\leq x}\sum_{d\mid(n,P_z)}\mu(d) = \sum_{d\mid P_z}\mu(d) \left[\frac{x}{d}\right], $$ where $P_z=\prod_{p\leq z}p$ where $p$ is prime, $\mu(d)$ is the ...
4
votes
1answer
91 views
+50

Applications of generating functions to number theory

I am familiar (at least at a cursory level) with the extensive role generating functions play in the theory of partitions. What are some other prominent applications of generating functions to number ...
0
votes
0answers
22 views

Find the highest LCM for n numbers in a range

I'm designing a component that takes a clock in (i.e. a periodic signal), and outputs a periodic signal with a lower frequency. To do so, I use two counters of different sizes. Here's an example, with ...
1
vote
1answer
51 views

Solve a system of diophantine equations

I have a problem with elemental number theory. I started with the expression $$ (a - \frac{1}{b})(b - \frac{1}{c})(c - \frac{1}{a}) $$ and task to find all natural $a,b,c$ so that the result of the ...
3
votes
0answers
38 views

Formula to round up to the next multiple not divisible by $2$ or $3$?

I want a formula that rounds up any integer to the next multiple of a given prime, which is not divisible by $2$ or $3$, so it is either $p$ or $5p \pmod{6p}$. The simplest formula is preferred. I've ...
1
vote
0answers
18 views

Show that the solutions lifted by Hensel's lemma, are the only solutions in $\mathbb{Z} / p^k \mathbb{Z}$

Suppose that $f(x) \in \mathbb{Z}[X]$ is a polynomial to which we can apply Hensel's Lemma. Suppose the set $A$ is the set of solutions of $f(x)$ in $\mathbb{Z} / p \mathbb{Z}$. I'd like to show that ...
0
votes
1answer
30 views

Prove that $a^4 \equiv 1 \bmod 5$ if $\space a \neq 5$

Prove that $a^4 \equiv 1 \bmod 5$ if$ \space a \neq 5$ I've tried showing this by induction. Clearly if $ a = 5$ then $ a \equiv 0 \bmod 5$ now if $a = 1$ then $a^4 - 1 = 0$ which is divisible by ...
1
vote
4answers
92 views

If $d=\gcd(a+b,a^2+b^2)$, with $\gcd(a,b)=1$, then $d=1$ or $2$

Suppose $\gcd(a,b)=1$. Let $d=\gcd(a+b,a^2+b^2)$. I want to prove that $d$ equals $1$ or $2$. I get that $d\mid2ab$ but I can't find a linear combination that will give me some help to use the fact ...
3
votes
0answers
79 views

diophantine equation $ |x^2-py^2|=\frac{p-1}{2} $

Prime $p\equiv3\pmod4$, then diophantine equation $$ |x^2-py^2|=\frac{p-1}{2} $$ has a solution in integers obviuosly, $x^2-py^2=-1$ has no solution in integers. Thanks!
5
votes
1answer
59 views

Problems that are made easy by using p-adic numbers

Does anybody know of elementary problems that can be be solved using the p-adics? Solutions are preferred.
1
vote
1answer
44 views

Integer solutions of the equation $a^{n+1}-(a+1)^n = 2001 $

I am doing number theory and I came across that question $a^{n+1}-(a+1)^n = 2001 $. Find the integer solutions and show that they are the only solution. I really tried hard but i am nowhere near ...
5
votes
1answer
79 views

For $n\ge4$, prove that $1!+2!+\cdots+n!$ cannot be the square of a positive integer

I'm trying to prove this by induction but seem to be getting nowhere.
1
vote
1answer
30 views

Proof on Divisibility of Binomial Coefficients

Prove that $\exists \ i$ $(0 \lt i \lt n)$ such that $$ n \nmid {n \choose i} $$ $\forall \ n$ such that $n \gt 0$ and $n$ is a composite Number.
1
vote
1answer
37 views

Concerning squarefree numbers with 2 primes and squarefrees with 3 primes.

If a squarefree with two primes is a 2-prime and a squarefree with three primes is a 3-prime is there an integer N such that the number of 2-primes less than N is equal to the number of 3-primes less ...
3
votes
1answer
37 views

$4|(p-1) \implies$ there is an element $x$ of order $ 4$ modulo $p$.?

"$p \equiv 1 \mod 4 \implies 4 \mid (p-1) \implies$ there is an element $x$ of order $4$ modulo $p$." I am having a difficult time understanding why this implies there is an element $x$ of order $4$. ...
0
votes
3answers
85 views

Testing If a Three/Four Digit Number is Prime or Not

Thank you for providing such great help. Thanks to math.stack site. I would like to know a good method to test any three/four digit number prime or not? I don't want to go any C or Java or any ...
0
votes
0answers
20 views

Convexity relation involving floor functions

In checking a convexity condition for a certain process indexed by the lattice, I have to verify that for any $\alpha$, $\beta$, and $t$ in $[0,1]$, the following holds for infinitely many $n$: $$ ...
2
votes
3answers
46 views

Prove that $(2m+1)^2 - 4(2n+1)$ can never be a perfect square where m, n are integers

I could prove it hit and trial method. But I was thinking to come up with a general and a more 'mathematically' correct method, but I did not reach anywhere. Thanks a lot for any help.
0
votes
1answer
47 views

infinite primes $p\equiv1\pmod n$ without cyclotomic polynomial

Without cyclotomic polynomial, is there an elementary proof of the following: for each integer $n>1$, there are infinitely many primes $p$ such that $p\equiv1\pmod n$ ? please don't refer to ...
2
votes
5answers
51 views

For any prime $p>3$ show that 3 divides $2p^2+1$

Does anyone know how to show this preferable without using modular For any prime $p>3$ show that 3 divides $2p^2+1$
2
votes
1answer
33 views

Concerning types of square-free numbers and comparing sizes of their subsets.

Call a square-free a 2-prime if it has exactly two prime divisors. Call a square-free a 3-prime if it has exactly three prime divisors,etc. Does there exist an integer N > 230 such that the number of ...
1
vote
2answers
42 views

Concerning types of square-free numbers.

Call a square-free number a 3-prime if it is the product of three primes. Similarly for 2-primes, 4-primes , 5-primes, etc. Are there two consecutive 3-primes with no 2-prime between them?Are there ...
0
votes
1answer
31 views

Concerning what is between two consecutive squares.

Is there a two-prime squarefree betweem any two consecutive squares?
2
votes
2answers
49 views

Describe all integers a for which the following system of congruences (with one unknown x) has integer solutions:

$$x\equiv a \pmod {100}$$ $$x\equiv a^2 \pmod {35}$$ $$x\equiv 3a-2 \pmod {49}$$ I'm trying to solve this system of congruences, but I'm only familiar with a method for solving when the mods are ...
2
votes
1answer
36 views

Problem on Number of Quadratic Residues

We have two primes $p,q$ and an integer $a$ such that $$\gcd(a,pq)=1$$ How to prove that for the following congruence $$x^2 \equiv a \mod pq$$ either there will be $4$ solutions or $zero$ solutions. ...
3
votes
2answers
90 views

elementary proof that infinite primes quadratic residue modulo $p$

$p \gt 2$ is a prime, then there are infinite primes $q$ such that $q$ is a quadratic residue modulo $p$. With Dirichlet's theorem on arithmetic progressions, the problem is easy! How about ...
4
votes
1answer
99 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 ...
0
votes
1answer
34 views

Evaluate smartly a function on a multiplication grid

I am asking myself the following question: Suppose one has a grid $G \in \mathbb{N}^{n\times n}$ where $g_{ij} = i\cdot j$, $i,j \leq n$. I would like to evaluate a function $f: G \to \mathbb{N}$. ...
0
votes
2answers
15 views

prove that $\forall k\in \mathbb Z$ if $(m,n)=1$(coprimes) then $(k,m,n)=1$

Prove that $\forall k\in \mathbb Z$ if $(m,n)=1$(coprimes) then $(k,m,n)=1$ I did this by contradiction: let $d=(k,m,n)$ so that $d\neq 1$; by definition of greatest common divisor $d|k, d|m, d|n$ ...
0
votes
5answers
108 views

Is this a true theorem? [closed]

I'm trying to prove the existence of the following theorem: If $n,p \in \mathbb{N}$, then $(p+1)^n = 1 \mod p$ Is this theorem true? I think it is, but I don't know how to prove it! Thanks!
0
votes
2answers
48 views

Proving divisibility tests using congruence relations [closed]

For a positive integer $N$ which has the decimal representation $$N=\sum_{k=0}^n a_k\cdot10^k $$ Prove that $$11\mid N \Longleftrightarrow 11\mid \sum_{k=0}^n(-1)^k a_k $$ using congruence ...
0
votes
1answer
41 views

Euclid's first theorem/ Euclid's lemma

How to prove that if $c$ divides $ab$ and $\operatorname{gcd}(a,c)=1$, then show that $c$ divides $b$. that means if $c|ab$ and $(a,c)=1 \implies c|b$.
1
vote
0answers
57 views

Proof Synopsis of Fermat's Last Theorem

I'm taking a introduction to higher math course now (mostly number theory) and our professor wants us to include two sentence proof synopses with our longer proofs. This got me thinking, What is a ...
11
votes
2answers
209 views

A different Harmonic series.

Let's call the following numbers than can be produced by playing with plus and minus: $$H_n'=\pm\frac{1}{1}\pm\frac{1}{2}\pm\frac{1}{3}\pm\cdots\pm\frac{1}{n}$$ "Harmonic kids" of $H_n$. We have a ...