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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3
votes
1answer
130 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 ...
4
votes
3answers
121 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 $x=2,y=3$ $$x=3,LHS=27-15+10=22$$ ...
1
vote
1answer
92 views

How many omegas are there in $\large f_{\varepsilon_0}$?

For a description look at fast growing hierarchy at wikipedia. $\large f_{\varepsilon_0}$ is not defined any more, it is a power tower of omegas, but how many omegas ? I found a defition $$\large ...
0
votes
2answers
449 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
2
votes
1answer
355 views

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 ...
0
votes
2answers
51 views

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 ...
7
votes
1answer
210 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.
0
votes
4answers
189 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 ...
11
votes
3answers
998 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. ...
1
vote
1answer
58 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
105 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
4answers
224 views

how do i prove that $17^n-12^n-24^n+19^n \equiv 0 \pmod{35}$

How do i prove that $17^n−12^n−24^n+19^n≡0(\mod35)$ for every possitive integer n. Can anyone give me a hint of how to start?
2
votes
0answers
54 views

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) ...
1
vote
1answer
59 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.
11
votes
5answers
447 views

What's the value of $n+\cfrac{n}{n+\cfrac{n}{n+\cfrac{n}{\vdots}}}$ for $n\in\mathbb{C}$?

Write $$\phi_n\stackrel{(1)}{=}n+\cfrac{n}{n+\cfrac{n}{\vdots}}$$ so that $\phi_n=n+\frac{n}{\phi_n},$ which gives $\phi_n=\frac{n\pm\sqrt{n^2+4n}}{2}.$ We know $\phi_1=\phi$, the Golden Ratio, so ...
1
vote
1answer
61 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 ...
1
vote
1answer
54 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
49 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
0answers
18 views

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 $g$?
33
votes
1answer
1k views

Is there a power of 2 that, written backward, is a power of 5?

In this note the famous mathematical physicists Freeman Dyson gives an example of a true statement that is impossible to prove. Or so he states. The statement is as follow: Numbers that are exact ...
0
votes
3answers
74 views

RSA decryption problem

(e,n) = (17,323), with ciphertext 185 First compute $\phi(323) = \phi(17*19) = 16*18 = 288$ In order to find the decryption exponent, we must solve 17*d = 1 mod 288 This is equal to $d = ...
1
vote
2answers
199 views

How do I compute Euler phi function efficiently for repeated prime factors?

In RSA decryption problems, you have to compute $\phi(n)$ and then sometimes $\phi(\phi(n))$ quickly. For example, I had to compute $\phi(2^5)$ for one particular problem and it seems to me (for ...
2
votes
0answers
56 views

Does Shor's algorithm work for noncommutitive or nonassociative algebras?

Shor's algorithm is said to solve the hidden subgroup problem for finite Abelian groups, at least according to the Wikipedia article I read. This means that it can be used to solve factoring or ...
1
vote
1answer
36 views

Greatest common divisor and exponent relationship

For a > 1 show that the gcd$(a^n - 1, a^m - 1) = a^{(m,n)} - 1$ What are some useful equalities that might help in proving this relationship? I believe the constrains for $m,n$ are all positive ...
2
votes
4answers
232 views

Find a value of $n$ that has exactly 32 divisors

I know that I could simply multiply the first $32$ primes together but is there some other way to ascertain the answer to this number theory problem?
2
votes
1answer
198 views

Is my conjecture correct? Any advice on how to solve this conjecture?

I was doing problem 6.3 from here. To make this less programming and more math oriented: GCDMany is equivalent to using Euclid's method (using mods and NOT ...
1
vote
0answers
187 views

Open Ball under the p-adic Norm

I'm trying to figure how, if it's even possible, to draw an open ball using the p-adic norm. My definition of the p-adic norm I'm using is: $ \lvert x \rvert_p $ = $p^{-ord_px}$ if $x \neq 0$ and ...
2
votes
0answers
80 views

Is factoring in other quadratic rings harder than factoring integers?

I've been wondering if factoring in quadratic rings that are unique factorization domains (principle ideal domains?) is more difficult than factoring integers. Today we can apply the general number ...
1
vote
3answers
76 views

The equation $b^2=a(a^2-1)$ has no rational solutions except obvious ones

I have problem with equation $b^2=a(a^2-1)$. How to show that except $(a,b)=(1, 0), (-1,0),(0,0)$, the equation hasn't any other rational solutions ? Editor's note. AFAICT the exercise is about ...
2
votes
1answer
56 views

$ax^2 + b$ and infinitely many primes: Does existence proof exist?

The question is on a subset of Bunyakovsky's Conjecture on an infinite number of primes existing in integer polynomials of degree higher than $1$. The conjecture itself is open. I have not been able ...
8
votes
1answer
116 views

GCD of $a^n + b^n$ and $c^n + d^n$

Prove or disprove that there does not exists any integers $a,b,c,d > 1$ such that $a,b,c,d$ are pairwise coprime, and $a^n + b^n$ and $c^n + d^n$ are also coprime for all integer $n > 1$. I ...
5
votes
1answer
208 views

Exponentials of rational numbers

Does there exist an $$0<x<1$$ such that $$\forall q \in \mathbb{Q^+}$$ $$q^x \in \mathbb{Q^+}$$
0
votes
2answers
68 views

The sum of two triangular numbers.

When triangular number is the square of an elementary formula is obtained. Sam got a couple of pieces, but I wonder how the formula looks opisyvayushaya sum of two triangular numbers is the square of ...
1
vote
1answer
68 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 ...
1
vote
1answer
41 views

How to establish $\sum_{d|n}d\phi(d)$

I am focusing on #5(b). I do not understand how they go from what I have to the answer. Those are r's at the end.
1
vote
4answers
55 views

Problem modulo $p$.

Let $p$ be a odd prime, prove that $1^p+2^p+...+(p-1)^p \equiv 0 \mod p$ I'm not sure how to do this, thanks.
1
vote
1answer
46 views

Prove that for $n\ge 2$, the n-th Lucas number is equal to $[a^n+1/2]$

Prove that for n greater than or equal to 2 the n-th Lucas number is equal to $[a^n+1/2]$. The brackets are the greatest integer function, $a = \frac{1+\sqrt5}{2}$. I get every kind of proof we ...
1
vote
1answer
683 views

Show that every nonzero integer has balanced ternary expansion?

show that every nonzero integer can be uniquely represented in the form $e_k3^k + e_{k-1}3^{k-1}+ … + e_13+e_0$ where $e_j= -1, 0, 1$ for $j = 0,1,2,…k$ and $e_k \neq 0$
0
votes
1answer
44 views

Details about Generalized Convolution (Number Theory - Apostol)

In "Introduction to analytic Number Theory" by Apostol there is chapter about generalized convolution. Let F denote a real or complex-valued function defined on the positive real axis such that ...
3
votes
0answers
62 views

The zeros of $2\,\xi(s)-1$. Is there anything known about the curves they lie on?

Take the well known integral: $$\displaystyle \pi^\frac{-s}{2}\,\Gamma\left(\frac{s}{2}\right)\, \zeta(s) =\int_1^{\infty} \left({x}^{\frac{s}{2}-1} + {x}^{\frac{-s}{2}-\frac12}\right)\,\psi(x)\, ...
0
votes
3answers
61 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 ...
4
votes
1answer
73 views

Generalized Pythagorean triples construction.

All primitive Pythagorean triples $(a, b, c) : \{ a^2 + b^2 = c^2 \} \wedge \{ a \equiv 0 \pmod{2} \}$ can be expressed in the form:$$\{ a = 2pq, b = p^2 - q^2, c = p^2 + q^2 \}$$ for positive ...
1
vote
1answer
84 views

If Cramér's is proved?

Harald Cramér proved that under this assumption that the Riemann hypothesis is true., the gap $g_n$ satisfies $$g_n = O(\sqrt{p_n} \ln p_n) ,$$ using the big O notation. Later, he conjectured that ...
2
votes
0answers
59 views

Calculate the number of times that $2$ appears in the prime-factorization of a given number?

Given a positive odd integer $K$ with a prime-factorization of $\prod\limits_{i=1}^{m}{P_i}^{N_i}$, is there a mathematical method for calculating the number of times that $2$ appears in the ...
5
votes
2answers
219 views

Is (the proof of) Fermat's last theorem completely, utterly, totally accepted like $3+4=7$?

If a mathematician would/does make use of Fermat's last theorem in a proof in a publication, would s/he still make use of some kind of caveat, like: "assuming Fermat's last theorem is true" or ...
1
vote
1answer
61 views

Ramification and roots of unity in complete discrete valuation rings.

Let $\mathcal{O}$ be a complete discrete valuation ring with algebraically closed residue field $k$ of characteristic $p>0$. Let $\pi\in \mathcal{O}$ generate the maximal ideal and suppose ...
0
votes
3answers
349 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
1answer
29 views

$\sum_{n=1}^{\infty}|a_{n} x^{n}| < \infty \implies \sum_{n=1}^{\infty} |a_{n}| (|x|^{n-1}+ |x^{n-2}y|+…+|y^{n-1}|) <\infty $?

Let $\{a_{n}\}_{n\in \mathbb N} \subset \mathbb C$ so that the series, $\sum_{n=1}^{\infty} a_{n} x^{n},$ converges absolutely for all $x\in \mathbb R$. Now, we let, $x, y \in \mathbb R$ with $x\neq ...
1
vote
3answers
50 views

$\sum_{n=1}^{\infty} a_{n}= \sum_{n=1}^{\infty}b_{n}$ and $\sum_{n=1}^{\infty} |a_{n}| < \infty \implies \sum_{n=1}^{\infty} |b_{n}| < \infty$?

Suppose that $\{a_{n}\}_{n\in \mathbb N}, \{b_{n}\}_{n\in \mathbb N} \subset \mathbb C$ such that both the series, $\sum_{n=1}^{\infty} a_{n}$ and $\sum_{n=1}^{\infty} b_{n}$ converges, and its sum ...
7
votes
1answer
87 views

$n^2$ divides $\phi(a^n-1)$ whenever $n$ is even and $a>2$

My problem is to show that $n^2$ divides $\phi(a^n-1)$ whenever $n$ is even and $a>2$. I have thought a solution but it is quite long and tedious. I wonder if anyone has a nice and clear ...