1
vote
1answer
31 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
31 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
14 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
29 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
74 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
28 views

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

Prime $p\equiv3\pmod4$, then the 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. How to attack this problem? Thanks ...
5
votes
1answer
53 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
42 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
75 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
28 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
36 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
35 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
70 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
19 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
45 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
45 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
49 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 $2p^2+1$
-1
votes
0answers
39 views

What are the last digits of 2 consecutive numbers? [closed]

What are the possible last digits of an integer of the form 2k(2k+1), when written in decimal?
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
41 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
30 views

Concerning what is between two consecutive squares.

Is there a two-prime squarefree betweem any two consecutive squares?
2
votes
2answers
48 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
35 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
82 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
96 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 ...
-2
votes
0answers
70 views

The Goldbach Conjecture [closed]

Could we consider the Goldbach Function as a proof of the conjecture since for any prime $\psi$ and for any real $\varphi$ ...
0
votes
0answers
25 views

The best generalization of a sequence [closed]

I have an equation, I will build sequences from this equation and I define a generalized equation and look for the best generalization of the sequences ! I will present an example and ask you if it is ...
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 $d|k, d|m, d|n$ ...
-1
votes
5answers
101 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
46 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
39 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
207 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 ...
0
votes
2answers
20 views

order of an integer related

I was reading a number theory text and this is when I encoutntered a line like this: "for $n=12$ , $\phi(12)=4$, yet there is no integer that is of order $4$ modulo $12$; indeed we find that ...
1
vote
0answers
27 views

Coprimality and division

I'm trying to understand 100% intuitively and rigorously ( at the same time ) almost all facts in basic number theory. I'm going really slow-paced and at the moment i didn't reach primes and unique ...
1
vote
1answer
47 views

Any odd > 1 is the average of three primes

I think that any odd integer is the average of three primes. My first question is if this is equivalent to some other conjecture/theorem in number theory (I suspect it is). But more importantly, I ...
0
votes
0answers
25 views

Logarithmic derivative of Riemann zeta, is this derivation correct?

Let matrix $T_2$ be defined below as the Dirichlet inverse of the Euler totient function as a function of the Greatest Common Divisor (GCD) of row index $n$ and column index $k$; $$T_2(n,k) = ...
6
votes
2answers
41 views

Number theory problem - contradiction

In an algebraic proof (for my problem it doesn't matter which proof) I have a special setting: $a,b,c \in \mathbb{Z}, \text{gcd}(a,c)=1,b<c \ \text{and} \ a \in \left\lbrace 1, \ldots , ...
1
vote
2answers
50 views

Let $t$ be a transcendental number. Prove that the set $\{(a+bt) \mid a,b \in \mathbb{Q}\}$ is not a number field.

Can I just pick a number in the set and then prove it's not constructible? Thx
0
votes
1answer
24 views

$lcm(a_{1},…,a_{n})=lcm(lcm(a_{1},…,a_{n-1}),a_{n})$

I tried to prove this by complete induction on $n$ but I am having problems in the inductive step: Suppose $$lcm(a_{1},...,a_{n})=lcm(lcm(a_{1},...,a_{n-1}),a_{n}) \forall k\le n\in \mathbb ...
1
vote
5answers
32 views

if $gcd(a,b)=gcd(a,b,c)$ then I need to prove that $ax+by=c$ has solution in $\mathbb Z$

if $gcd(a,b)=gcd(a,b,c)$ then I need to prove that $ax+by=c$ has solution in $\mathbb Z$ that is: $gcd(a,b)|c$ but how can I prove it with the given hypothesis?
0
votes
0answers
24 views

prove that $ax+by=b+c$ has solution in $\mathbb Z$ if and only if $ax+by=c$ has solution in $\mathbb Z$

Let $a,b,c,d\in \mathbb Z$ prove that: a)$ax+by=b+c$ has solution in $\mathbb Z$ if and only if $ax+by=c$ has solution in $\mathbb Z$ b)$ax+by=c$ has solution in $\mathbb Z$ if and only if ...
0
votes
1answer
29 views

proof of $lcm(a_{1},…,a_{n})=lcm(lcm(a_{1},…,a_{n-1}),a_{n})$

prove that $lcm(a_{1},...,a_{n})=lcm(lcm(a_{1},...,a_{n-1}),a_{n})$ ($n\in \mathbb N$) By complete induction on n: (n=1,2 are trivial) for n=3: let $c=lcm(lcm(a_{1},a_{2}),a_{3})$ and ...
3
votes
1answer
81 views

Proof of $(ma+ nb, mn)=(a,n)(b,m)$

Let $a,b,m,n \in \mathbb Z$. If $(m,n)=1$ ( $m,n$ are coprime integers) prove that $(ma+ nb, mn)=(a,n)(b,m)$ I started the proof like this: Let $c,d,e$ be the greatest common divisors of ...
0
votes
2answers
43 views

Is this solvable: $x^{2}\equiv5\pmod{229}$?

$$x^{2}\equiv5\pmod{229}.$$ Using Legendre symbol, $(\frac{5}{229})(\frac{229}{5})=(-1)^{\frac{4}{2}*\frac{228}{2}}=(-1)^{2*114}=(-1)^{228}=1.$ Hence, 5 is a quadratic residue $mod(229)$ if 229 is a ...
0
votes
1answer
21 views

gcd , lcm problem with divisibility application

How should i prove that $ab|(a,b)[a,b]$ ? Here $(a,b)=gcd(a,b)$ and $[a,b]=lcm(a,b)$. I tried and got an answer as $\frac{ab}{(a,b)}|[a,b]$. then i can also proceed as $[a,b]=(a,b)^2\frac{k}{ab}$. ...
1
vote
1answer
47 views

How to show $(1^2)(3^2)(5^2)…((p-2)^2)=(-1)^{(p+1)/2}$

I want to show the above problem using Wilson's theorem, which I know is $(p-1)!\equiv(-1)$ mod p. If I start with this I get $1\dot{}2\dot{}3\dot{}...\dot{}(p-1)\equiv(-1)$ mod p, but I don't know ...
1
vote
0answers
64 views

How to prove $\pi ^{3}$ is not constructible from the fact that $\pi $ is not constructible?

I know how to do this for $\sqrt[3]{\pi }$: First suppose it is constructible and then you just set it equal to $x_{0}=\sqrt[3]{\pi }$ and take the third power of both sides. Then you get ...
6
votes
1answer
103 views

Find all natural numbers such that $\sum_{k=1}^{n} \frac{n^k}{k!}$ is an integer

Find all natural numbers such that $\sum_{k=1}^{n} \frac{n^k}{k!}$ is an integer. I've tried to bring all fractions under commmon denominator and it didn't helped me much. With guessing I find out ...