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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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 ...
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62 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 ...
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32 views

Triangular numbers for numbers.

Interestingly for triangular numbers: $X(X+1)+Y(Y+1)=Z(Z+1)+a$ $a$ - this number is determined by the condition of the problem. Are all numbers equation has a solution? And what kind of formula in ...
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30 views

Every integer greater than 1 is divisible by at least one prime. Can anyone please express this in logical notation

Every integer greater than 1 is divisible by at least one prime. Can anyone please express this in logical notation
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Stuck in quadratic forms and discriminats problem

So I'm stuck in a pretty easy question about discriminants and quadratic forms of equations. I have already proved one side of the problem: we suppose that $x_0, y_0$ are the solutions to the ...
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integral roots for $f(x) = 41$ if $f(x) = 37$ has 5 distinct integral roots.

Given a polynomial $f(x)$ with integral coefficients and $f(x) = 37$ has 5 distinct integral roots, find the number of integral roots of $f(x) = 41$? My Approach: Say $f(x) = ...
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How to convert a integral into the another?

There is a function here: http://functions.wolfram.com/NumberTheoryFunctions/PrimePi/21/01/01/0001/ How to convert it into the answer for indefinite integral $\int \pi(x) dx$ where pi(x) stand for ...
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75 views

How find this $x^3-5x+10=2^y$

let $x,y$ is positive integer,and such $$x^3-5x+10=2^y$$ find all $x,y$. since $$x=1\Longrightarrow 1^3-5+10=6$$ can't $$x=2,2^3-5\cdot 2+10=8=2^3$$ so $x=2,y=3$ $$x=3,LHS=27-15+10=22$$ ...
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28 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 ...
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Given the factors of $N$, is there a method for computing the factors of $N-1$ or $N+1$?

Given the prime factorization of $N$, is there a known method for computing the prime factorization of $N-1$ or $N+1$, which is more efficient than the best known method for doing that without it? I ...
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1answer
77 views

Firoozbakht's conjecture solution?

Not so much an question as adding another level to the same question as Ratio of logarithmic primes. (See answers, same as here.) The Firoozbakht's conjecture (1982) is equal to: $$(p_{n+1})^{n} ...
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1answer
17 views

count of Ordered Pairs such that their product is less than a number

I need a mathematical formulation for count of total ordered sets s.t. the product of two elements is less than an number, say n.. count{(i,j)}, s.t.i*j<=n
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1answer
39 views

Farey Sequence Vector Orthogonality Relation Question

Take the Farey sequence $\mathcal{F}_n$ for $n=39$ with values $a_m\in \mathcal{F}_n$ and put them into a vector $$ \vec v_k=\biggr(\exp(2\pi i k a_m)\biggr)_m $$ Since Merten's function for $n=39$ ...
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90 views

Is the sum of roots of unity always a real multiple of a root of unity?

I can see this is true for the sum of two roots of unity with some basic trigonometry (the resulting argument is the half the sum of the original arguments, and so must also be a rational multiple of ...
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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 ...
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25 views

Is there any other integral of a special function that is undetermined?

Is there any other integral of a special function that is undetermined but yet the special function itself is continous? eg. the integral of the prime number counting function is undetermined By ...
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1answer
30 views

Comparing coefficients in finite field

We start with the wrong proof of the following theorem: $p| \binom{p}{k}$ for a prime $p$ and $0<k<p.$ Proof: $(1+x)^p \equiv 1+x \equiv 1+x^p \pmod{p}$ by Fermat's little theorem. Comparing ...
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1answer
27 views

Are we only knowing prime counting function's property but not its infinite expansion?

Are we only knowing prime counting function's asymptotic property but not its infinite expansion or even people could saying that there are no infinite series for the function? If yes, what are some ...
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2answers
12 views

Proof that in any base $b$, the result of multipling two numbers of $k$ digits, doesn't recuire more than $2k$ digits

The proof that I came up whit is: Let, $c$ be $b^0 r_0+ b^1 r_1+b^2 r_2+...+b^k r_k$ and $d = b^0 r_0'+ b^1 r_1'+b^2 r_2'+...+b^k r_k'$ then multipling both: $$(b^0 r_0+ b^1 r_1+b^2 r_2+...+b^k ...
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1answer
25 views

Calculate modulo of large numbers

I have $2^{2^n}+1$ and i want to calculate ($(2^{2^{^n}} +1 )\mod 19$). How can i do it if for example i choose $n = 19$. Can i use Fermat's Little Theorem?
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how do i prove that $17^n-12^n-24^n+19^n \equiv 0 \pmod{35}$

How do i prove that 17n−12n−24n+19n≡0(mod35) for every possitive integer n. Can anyone give me a hint of how to start?
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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 ...
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6 views

When does $f_{\omega+1}$ catch up the $G_n$-sequence?

Which is the minimal number k, so that $f_{\omega+1}(n) > G_n$ is true for all $n\ge k$ ? For the definition of $f_{\omega+1}$ look at wikipedia fast growing hierarchy $G_n$ is defined by ...
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28 views

No prime number divides one

I was reading Euclid's theorem and came accross this affirmation but no prime number divides 1 Is there any mathematical proof or is it an axiom of number theory ? Can this affirmation be ...
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How many omegas are there in $\large f_{\epsilon_0}$?

For a description look at fast growing hierarchy at wikipedia. $\large f_{\epsilon_0}$ is not defined any more, it is a power tower of omegas, but how many omegas ? I found a defition $$\large ...
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2answers
45 views

Find three numbers given their sum, product and sum of their squares

Given three unknown positive integers. Is it possible to find the three numbers if we are given their Sum->(a+b+c) = X Product-> (abc) = Y Sum of Squares-> (a^2 + b^2 + c^2) = Z
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Twin Prime conjecture current status

Can someone help me with a link to read about the status of the Twin Prime conjecture. I have browse on the internet and have read some articles but still I have no clue of the updated status of Twin ...
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Solving Linear Congruences $ax+c = b \pmod{m}$

I am facing this problem i know how to solve $ax = b \pmod{m}$. For example $16x = 52 \pmod{52}$: I know that the result is $0,13,26,39$ or $13k$ but what is the solution for $16x + 48 = 52 ...
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345 views

Prove that $\sum\limits_{i=0}^{k} p^{2i}$ ($p$ is prime) is never a perfect square

Prove that $$ \sum_{i=0}^{k} p^{2i} $$ where $k > 0$ and $p$ is an arbitrary prime, is never a perfect square. I think you can prove it by letting $q = \sum\limits_{i=0}^k a_ip^i$, then expanding ...
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268 views

Lower bounds for the partition function

In this question we consider the partition function $p(n)$ - that is, the number of ways to express $n$ as a sum of positive integers. One easy exercise is to show that $$ p(n) \geq 2^{\lfloor ...
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945 views

Is this already an equation/law that has been found?

So I was messing around with some numbers today and I have found a way to quickly add summations (probably not the first one to discover it but...) this only works when you start at 1 (i.e. ...
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1answer
43 views

The difference set $D(\mathbb Z^*_n)$ of $\mathbb Z_n^*$

I wish to ask whether $D(\mathbb Z^*_n)=\mathbb Z_n^+$ given $n$ is odd. This is equivalent to proving that: For every $l\in\mathbb Z^+_n$, the set $l+\mathbb Z^*_n=\{a+l:a\in\mathbb ...
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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 ...
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79 views

A question on consecutive prime numbers

Prime numbers: 2 3 5 7 11 13 17 19 23 29 .... Difference between to consecutive primes: 1 2 2 4 2 4 2 4 6 .... We know that there are infinite prime numbers. This is Ok. But does the difference ...
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Conjecture: the sequence of sums of all consecutive primes contains an infinite number of primes

Starting from 2, the sequence of sums of all consecutive primes is: $$\begin{array}{lcl}2 &=& 2\\ 2+3 &=& 5 \\ 2+3+5 &=& 10 \\ 2+3+5+7 &=& 17 \\ ...
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37 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.
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4answers
58 views

Prove that (integer + non-integer) never equals an integer.

My question is how do you prove that given an integer $x$ and a number $y$, the only way for $x + y$ to be an integer is if $y$ is also an integer. I can see how to prove by induction that an integer ...
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2answers
417 views

Riemann's zeta as a continued fraction over prime numbers.

Riemann's zeta function is a function with many faces, I mean representations. I recently derived this one, bellow, as a continued fraction over prime numbers. $$ \zeta(s)=1 ...
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1answer
40 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 ...
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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.
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Is there an infinite number of primes of the form (5^n)-2 and/or (5^n)+2?

I would like to know if there is a proof of this, that shows whether either one or both of the expressions: $5^n-2$ and $5^n+2$ will equal a prime number indefinitely. (Is the set of values for (n) ...
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1answer
27 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.
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Show that there is no integer n with $\phi(n)$ = 14

I did the following proof and I was wondering if its valid. It feels wrong because I didn't actually test the case when purportedly n is not prime, but please feel free to correct me. Assume there ...
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98 views

An intutive way to think about odd and even numbers. [on hold]

What is an intuitive way to think about odd and even numbers? And about divisibility also...
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1answer
19 views

Concerning integer intervals

Call an integer square-in if it is not square-free or a square. Can two consecutive square-in numbers have a gap of <8 integers between them?Exactly one of these integers in this 'gap' being a ...
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1answer
41 views

Determining if a number is an nth root

I am working on a proof that depends on if an adversary can determine if a number is an $nth$ power for some large prime $p$. My intuition tells me that for a sufficiently large value of $n$ this is ...
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1answer
86 views

Solutions to $x^p+y^q = z^r$

Is there any $(p,q,r)$ with $\gcd(p,q,r) = 1$ and $\frac{1}{p}+\frac{1}{q}+\frac{1}{r} < 1$ for which we know that the only integer solutions (not necessarily primitive) to the equation ...
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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 $4$. ...
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Average Orders and Convolutions

If I know the average order of an arithmetic function $f=I*g$, where $I$ is the identity function defined by $I(n)=n$, is there a way to find the average order of $g$?