Questions on more advanced topics of number theory, such as quadratic residues, primitive roots, prime numbers, non-linear Diophantine equations, etc. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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4
votes
1answer
72 views

Integer part of $e^x$

Using wolfram alpha gives that $\lfloor e^{2015}\rfloor \equiv 1 \pmod 4$. Is it possible to prove this by manual ? Thank you.
-1
votes
0answers
14 views

Finite sequence of positive integers with $a_k \equiv k (\mod m)$

Is it true that for every strictly increasing finite sequence of positive integers $(a_1, a_2, \ldots, a_n)$, $a_1<a_2<\ldots<a_n$, there exists a positive integer $m>1$ such that $a_k ...
0
votes
1answer
27 views

Understanding Erdös Discrepancy Conjecture for random series of -1, 1…

In today's newspaper NRC from The Netherlands a two-page article was dedicated to computer-aided mathematical proofs (sorry, paid link and in Dutch). Interesting in itself, but as an example it used ...
-1
votes
1answer
24 views

Probability and Data Integrity

This question is about probability and Security (i.e. data integrity). The scenario I am going to explain is a client-server case where the server may modify the client's data. We define a field ...
1
vote
2answers
19 views

To simplify $\sum\limits_{k|(2m, 2n), k\nmid m} \varphi(k)$

Consider the Euler phi-function $\varphi(n)$ of $n\in \mathbb N$ as the cardinality of $\{1\leq r\leq n: (r, n)=1\}$. I am willing to simplify $$S:=\sum\limits_{k|(2m, 2n), k\nmid m} \varphi(k).$$ ...
2
votes
2answers
46 views

Sum of (only certain) prime reciprocicals

It is well known that $$ \sum_{p\ is\ prime}\frac1{p}$$ diverges. Is there a simple proof that $$ \sum_{p\equiv 1\pmod 4}\frac1{p}$$ and $$ \sum_{p\equiv 3\pmod 4}\frac1{p}$$ also diverge? (p ...
-3
votes
3answers
42 views

Trinomial Equation

$$x^n + x^m = 1$$Given $n,m$ are two positive distinct integers, prove that $x$ is irrational number which can be expressed only in terms of $n,m$ (Assume $n$ is greater than $m$)
-1
votes
0answers
57 views

Can anyone show me a solution where we solve a quintic and find its 5 roots properly using elliptic functions? [duplicate]

Kindly use this quintic equation:- $$ 5x^5 + 4x^4 + 3x^3 + 2x^2 + x^1 = 5 $$ Find roots of the above equation using elliptic function? I need this solution very badly?
5
votes
2answers
62 views

$n-\frac{n(n^2-1)}{2!}+\frac{n(n^2-1)(n^2-4)}{2!3!}-\frac{n(n^2-1)(n^2-4)(n^2-9)}{3!4!}+…$

If $n\in \mathbb{N}$, and $n-\dfrac{n(n^2-1)}{2!}+\dfrac{n(n^2-1)(n^2-4)}{2!3!}-\dfrac{n(n^2-1)(n^2-4)(n^2-9)}{3!4!}+\dots$ $=s_1$ when $n$ is even and $s_2$ when $n$ is odd then prove that ...
0
votes
3answers
40 views

Proof verification - $\sqrt[3]{2}$ is irrational

This might be a duplicate, but I have not found one on this site. This is my proof and I was wondering if this proof depends on it's conclusion. Here it is: Assume $\sqrt[3]{2}$ is rational. Then ...
3
votes
2answers
46 views

coefficients that make $p(x)=(a_1x+b_1)^3+(a_2x+b_2)^3+(a_3x+b_3)^3+(a_4x+b_4)^3-x$ a constant

Find integers $a_i$ and $b_i$, $i=1,2,3,4$, such that $p(x)$ is a constant function: $p(x)=(a_1x+b_1)^3+(a_2x+b_2)^3+(a_3x+b_3)^3+(a_4x+b_4)^3-x$ I don't even know if such coefficients exist or not. ...
0
votes
1answer
9 views

calculating differences between entries in a table using each entry or skipping entries after rounding

Entry two of a table is entry one plus or minus some number. Each entry is determined by adding or subtracting some value to/from the previous entry. The value varies. The entries are composed of ...
5
votes
5answers
78 views

A trick for calculating $n^6$ that I don't understand

I was doing a math exercise and it asked to find what are the possible units digits of $n^6$ knowing that $n\in\mathbb Z$. The solution said that because we are concerned only with finding what the ...
2
votes
0answers
36 views

How to calculate the $n$ prime from $\pi (n)$?

Assume we had an exact formula for $\pi (n)$, how could we get from that formula an exact expression for the $n$th prime? I tried looking at approximations we have of $\pi (n)$ like $\frac {n}{\ln ...
0
votes
0answers
38 views

How to compute the correct residue without change of modulus?

Suppose $F=1857$, $G=2017$, and we want to compute $ FG \equiv 1606$ mod $(2^{11} -1)$. Also let $t =2^4$, then $F = 1+4t+7t^2$ and $G = 1+14t+7t^2$, where the most significant digits are bounded ...
0
votes
1answer
33 views

For $ \frac{n(n-1)}{2}<i \leq \frac{ n(n+1)}{2} $, why is $\text{round}(\sqrt{2i})=n$

I was doing Problem 67 of Project Euler, which involves number triangle which has similar structure to Pascal's triangle (the nth row contains n numbers). While thinking of a way to represent the ...
1
vote
1answer
43 views

Derivation of Perron's formula

I tried to derive Perron's formula, but got really screwed up. I know of other ways to derive it, but I'm not quite sure why this way isn't working. I would appreciate some pointers on where I'm going ...
7
votes
0answers
81 views

Primes $p$ for which $2p \pm 1$ are also primes

Out of curiosity and trouble sleeping, I decided to look at the distribution of primes $p$ for which $2p \pm 1$ are also primes. I looked at the first 25,910,000 primes and counted the number of ...
2
votes
0answers
34 views

Closure of Set of Fractions with Lowest Terms Condition

Suppose I have a set of rational numbers where elements have denominators are odd and numerators and denominators are co-prime. I need to show that the set is closed under addition. It is clear that ...
0
votes
0answers
26 views

Test on representing prime using eight variate quadratic form

Assume $f_i(x_1,\dots,x_8)$ are linear forms in $\Bbb Z[x_1,\dots,x_8]$ where $i\in\{1,\dots,m\}$. Assume $c_i\in\Bbb Q$ are constants where $i\in\{1,\dots,m\}$. Consider quadratic form ...
1
vote
0answers
28 views

Rate of large composite numbers, which are strong probable prime to the bases $2,3$ and $5$

Here http://primes.utm.edu/glossary/xpage/StrongPRP.html is the definition and some useful informations about strong probable primes. For higher numbers, lets say near $10^{50}$, strong probable ...
-2
votes
2answers
30 views

Rank of an $m$ by $n$ matrix?

Can anyone state, in plain English, how to find the rank of an $m$ by $n$ matrix? Is it necessary to perform Gaussian elimination first, or translate it into upper triangle form (or however it is ...
0
votes
0answers
79 views

Is there more than one looping sequence in the Collatz conjecture? [on hold]

Is it known whether there is more than one loop in the Collatz conjecture? Following advice and warnings on meta, I try below to claim that there is only one looping sequence in all the sequences ...
1
vote
3answers
52 views

Is there an error in this GRE question?

I was doing a Manhattan GRE practice exam and I was sure I had cracked the twist in this one question... only to find in that there was no twist (apparently). Here is what I know: $$ \sqrt a = \pm b ...
1
vote
1answer
22 views

Proof of a complete residue system without using congruences

I am taking a elementary class on number theory this semester, and among the exercises from the third lecture there is this one: For $m > 1$ and $gcd(m, a) = 1$, show that the remainders from the ...
0
votes
1answer
18 views

Count of solutions to matrix equations

Given these modular equations: $$a_{1,1} x_1 + a_{1,2} x_2 + \cdots + a_{1,n} x_n = b_1 \bmod p $$ $$a_{2,1} x_1 + a_{2,2} x_2 + \cdots + a_{2,n} x_n = b_2 \bmod p $$ $$\vdots$$ $$a_{m,1} x_1 + ...
15
votes
2answers
1k views

Who is the “father of number theory”?

I noticed that some sources state Fermat as the father of modern number theory while others say Gauss. I am trying to start a paper on the history of number theory for a presentation, but I cannot ...
1
vote
1answer
19 views

Proof Verification - $\exists a \in S (a\ge S_a)$

I wanted to prove that $\exists a \in S (a\ge S_a)$ where $S$ is an finite set of real numbers with order $n$ and $S_a$ is the average of the set. This is my proof so far: Assume $a_i = a_k, i,k ...
0
votes
2answers
55 views

Fermat's Last Theorem - Variation with arithmetically descending exponents

Are there solution(s) to the following variant of Fermat's Last Theorem in the positive integers? $$ a^n + b^{n-i} = c^{n-2i} $$ I haven't been able to identify any trivial solutions. To my ...
1
vote
3answers
55 views

Proof that if $\gcd(m,n) = 1$, then $\gcd(m+n,mn ) = 1$. [duplicate]

I need help with this excercise. If $\gcd(m,n) = 1$, then $\gcd(m+n,mn ) = 1$. I don't know how to prove this, I know the definition of $\gcd$ but I can't prove it.
4
votes
1answer
98 views
+50

Connection between Dirichlet series and integration?

For quiet sometime I've been working on an idea of mine: Basis We define the following basis: $$ A_n= ( \underbrace{00000...}_{n-1\text{ times}} 1 )^T $$ Hence, $$ A_1 =(111111 ... )^T $$ $$ A_2 ...
0
votes
1answer
33 views

Is $p_n \sim \frac{5}{4}n\log(n) + \frac{1}{2}n + \frac{(p_1+\ldots+p_{n-1})}{n-1}$ a good approximation for the $n^\text{th}$ prime?

If you plot the following function $$f(n) = \left|\frac{(p_1+\ldots+p_{n})}{n} - \frac{(p_1+\ldots+p_{n-1})}{n-1}\right|$$ you get a graph that is similar to $$f(x) = \frac{5}{4}\log(x) + ...
0
votes
0answers
24 views

Implementing FizzBuzz game

I need to build an electrical-circuit for the FizzBuzz game. There's a signal, called next which increment the current number by one. The rules are simple - You ...
3
votes
0answers
53 views

Trisectible Angle

How do we prove that a triangle with sides $(one, x, y)$, where $x$ is any constructible length from one to three at the elliptic curve $$y^2 = x^3 -x^2 -x +1$$then the triangle possess at least ...
6
votes
0answers
62 views

Fermat's Last Theorem: A natural extension

It is well known that there are no solutions to $$a_1^n+a_2^n=b^n$$ for $a_1,a_2,b\in\mathbb{Z}^+$ and $n>2$. Is it then true that there are no solutions to $$a_1^n+a_2^n+\cdots+a_m^n=b^n$$ ...
8
votes
0answers
73 views

Can anyone improve on this work and find a closed form of $\zeta(3)$?

This was something I and another user came across independently, although he decided to post it on reddit. So while its already online, let me reproduce it here with the hope that someone will be able ...
3
votes
1answer
27 views

Was this arithmetic Möbius/Mangoldt function ever used for something?

Let $n=\prod_k p_k^{c_k}$, with $p_k \in \mathbb P$ and $$ A(n)=\sum_{d|n} \mu(d)\Lambda(d), $$ with the $\mu$ Möbius function, which has values in {−1, 0, 1} depending on the factorization of n ...
5
votes
5answers
140 views

What is $\limsup_{n\to\infty} \frac{p_{n+1}}{p_n}$?

Let $(p_n)_{n\in\mathbb N}$ be the strictly increasing sequence of all primes. I'm wondering what $$S:=\limsup_{n\to\infty} \frac{p_{n+1}}{p_n}$$ is. Is the result already known? By Bertrand's ...
1
vote
2answers
27 views

Proof - Uniqueness part of unique factorization theorem

The uniqueness part of the unique factorization theorem for integers says that given any integer $n$, if $n=p_1p_2 \ldots p_r=q_1q_2 \ldots q_s$ for some positive integers $r$ and $s$ and prime ...
0
votes
0answers
27 views

Limit of an euler product

Before I can ask my question, I have to state a couple of definitions. Let $f$ be a multiplicative function and let $$ D_f(s) = \sum_1^{\infty} \frac{f(n)}{n^s}, $$ and define $\Lambda_f(n)$ as ...
0
votes
0answers
37 views

Does this set contain these numbers?

How would I go about proving whether or not every number $n=k^8$ is included in the set of all numbers $m=k^4$ ($n$ and $k$ are integers in both cases)?
1
vote
2answers
33 views

Lucas's proof of a special case of Beal's conjecture

While studying the properties of a certain elliptic curve, I came across the equation $x^4+y^4=z^3$. There is no solution of this equation in relatively prime integers, and this is a special case of ...
3
votes
4answers
125 views

How to show that $2\times 10^{18}<20!<3 \times 10^{18}$ without calculator? [on hold]

I want to find the first digit of $20!$ By calculator $20! = 2.43290200817664 \times 10^{18}$. So I want to show that $2\times 10^{18}<20!<3 \times 10^{18}$ Thank you.
3
votes
2answers
84 views

Consecutive squarefree numbers of 5 prime factors each, mostly small

The sequence of numbers 49297533, 49297534, and 49297535 is notable, because the factorizations of these numbers are each of the form $a^1 \cdot b^1 \cdot c^1 \cdot d^1 \cdot e^1$, where $\{a\ldots ...
5
votes
1answer
60 views

How to prove$\displaystyle \sum_{i=0}^{k}(-1)^i\binom{n}{k-i}\binom{n+i-1}{i}=0$

I saw a combinatorial identity when i study linear-algebra, But the author didn't explain how to get it. $\displaystyle \sum_{i=0}^{k}(-1)^i\binom{n}{k-i}\binom{n+i-1}{i}=0$ I tried $n=10$ or ...
2
votes
1answer
37 views

Likelihood at least 2 out of $n$ numbers are visible to each other in $\mathbb{Z}^n$

Two points in $ \mathbb{Z}^n $ are said to be visible to each other, if they can be connected by a straight line, which doesn't intersect any points of $ \mathbb{Z}^n $ In Apostol's book "An ...
0
votes
0answers
24 views

On some factorial inequalities

Denote $P_n$ to be product of primes at most $n$. What is the minimum value of $m$ such that $P_m\geq P_n^2$? What is the minimum value of $m$ such that $m!\geq n!^2$? What is the minimum value of ...
1
vote
5answers
70 views

there does not exist a perfect square of the form $7\ell+3$

I have been trying to prove that there does not exist a perfect square of the form $7\ell+3$. I've tried using $n$ as even or odd, and I'm getting stuck. Can someone put me on the path? Is this an ...
1
vote
0answers
16 views

Why does the uniqueness theorem for Dirichlet series hold for the infinite sums, while obviously not for partial sums?

I asked in a previous question whether a function, $a_n$, is unique to $F(s)$ for any Dirichlet function defined by the following $$F(s)=\sum_{n=1}^\infty{\frac{a_n}{n^s}}.$$ Its uniqueness property ...
1
vote
0answers
17 views

What are the asymptotic considerations in the following?

The following is from this paper that discusses polynomials and classic number theory functions. The proof of theorem 1.3 has a final statement saying that $R$ must be null because we arrive at ...