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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1answer
91 views

How to solve special type of Diophantine equation

I am so excited to learn finding integer solutions of the equation $x^2 -y^5 = x-y$. I just found few solutions by plugging various integers in place of $x$ and $y$. But, I need a permanent method or ...
1
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3answers
26 views

Minimize given LCM

Find the smallest possible value of $n_1+n_2+\cdots+n_k$ such that $LCM(n_1,n_2,\ldots,n_k)=(2^2)(3^3)(5^5)$. Note that $k$ is not fixed. I know the answer should be $k=3$, $n_1=2^2$, $n_2=3^3$, and ...
1
vote
1answer
88 views

regularization of sum $n \ln(n)$

I was testing out a few summation using my previous descriped methodes when i found an error in my reasoning. I'm really hoping someone could help me out. The function which i was evaluating was ...
1
vote
1answer
43 views

Induction Proof - Primes and Euclid's Lemma

I'm having some trouble with this proof. Here's the question: Use mathematical induction and Euclid's Lemma to prove that for all positive integers $s$, if $p$ and $q_1, q_2, \dotsc, q_s$ are prime ...
2
votes
1answer
52 views

Which power of an integer matrix is identity modulo $p^\alpha$?

I've read this question about identity power of an integer matrix. But how about power of a matrix modulo $p^\alpha$. $$A^m \equiv I \pmod{p^\alpha} $$ How can I find the minimal $m$ that the above ...
5
votes
5answers
87 views

Decomposing an integer into primes raised to different powers

The number $711000000$ can be written as $79^1 \times 2^6 \times 3^2 \times 5^6$. How are these numbers found? I guess the more general question is - given $n \in \mathbb Z $, how can you ...
2
votes
2answers
152 views

Solving the equation $ x^2-7y^2=-3 $ over integers

I'd like to solve the following Pell equation: $$ x^2-7y^2=-3 $$ Where $x$ and $y$ are integers. I applied the usual procedure, which avoids continued fractions: The two minimal positive integer ...
4
votes
2answers
69 views

Ordering of natural numbers

Show that it is possible to arrange the numbers 1, 2, . . . , n in a row so that the average of any two of these numbers never appears between them. Hint: Show that it suffices to prove this fact ...
-2
votes
1answer
45 views

Fi Binary Number [closed]

A Fi-binary number is a number that contains only 0 and 1. It does not contain any leading 0. And also it does not contain 2 consecutive 1. The first few such number are 1, 10, 100, 101, 1000, 1001, ...
7
votes
2answers
125 views

Arithmetic Derivatives: Arithmetic Logarithmic Derivative Problem

In Calculus, whenever we see a constant and want to take the derivative of it, it always is 0. However in Number Theory, we have something called the arithmetic derivative in which we can ...
7
votes
1answer
47 views

Orbits of the $\text{SL}(n,\mathcal{O}_K)$-action on $\mathbb{P}^{n-1}(K)$ for a number field $K$.

I was reading some notes of Keith Conrad where he proves that the number of orbits of the $\text{SL}(2,\mathcal{O}_K)$-action on $\mathbb{P}^{1}(K)$ for a number field $K$ is precisely the class ...
3
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1answer
44 views

Chiese Remainder Therorem : No solutions

I have asked a similar question before on Chinese Remainder Theorem. Now concepts are getting clear. Thinking of a possible case where there are no solutions. Suppose the question is ...
0
votes
1answer
60 views

Bowers array notation : $f_{\omega^\omega}(n)\approx [n,…,n]$ ($n$ times)

I learnt at this site that $$\large f_{\omega^\omega}(n)\approx \underbrace{[n,...,n]}_{n\ n's}$$ For a simular approximation $$\large f_{\omega^2}(n)\approx \underbrace{n\rightarrow ...
2
votes
1answer
80 views

X raised to power-X raised to power-3 equals to 3.

The question is what are the possible values of $x$ when we have $$x^{x^3} = 3$$ (that is $x^3$ in the exponent itself and not $x*3$). I solved one answer by guessing that $x = \sqrt[3]3$. My work ...
4
votes
1answer
84 views

Integer solutions to $x^2=2y^4+1$.

Find all integer solutions to $x^2=2y^4+1$. What I tried The only solutions I got are $(\pm 1 ,0)$, I rewrote the question as : is $a_{n}$ a perfect square for $n>0$ were $$a_0=0,\quad ...
5
votes
1answer
57 views

The Sieve of Eratosthenes as sum of square waves

I wrote this equation, that is a way to represent the Sieve of Eratosthenes: $-1+\sum\limits_{i=2}^{\infty} ( 2 \left \lfloor \frac {x}{i} \right \rfloor - \left \lfloor \frac {2x}{i} \right \rfloor ...
0
votes
1answer
18 views

Hilbert Symbol over $\mathbb{R}$ (bilinearity)

Let $\mathbb{R}$ be the field of the reals and let $a,b,c \in \mathbb{R}^{\times}$. As you probably know, the Hilbert symbol over any field $K$ is defined as: $$(\frac{a,b}{K}) = 1 \text{ if } \exists ...
0
votes
0answers
31 views

Unique extension of the absolute value

Let $(K,u)$ be a complete valued field, $u$ be its discrete absolute value (corresponds to a discrete valuation on $K$), then: ($\ast)$ Let $E/K$ is a finite separable field extension, then the ...
2
votes
2answers
62 views

percentage of integers such that $n^4 \pmod{16} \equiv 1$?

How do I find the percentage of numbers $n$ in the list $1^4, 2^4, ... 1000^4$ such that $n \pmod{16} \equiv 1$? I know that for any $x$, if $x \pmod{16} \equiv 1$, then $x^n \pmod{16} \equiv 1$, so I ...
1
vote
2answers
128 views

Primality of $2^{255}-19$

I need a test for primality that I apply to $2^{255}-19$ (which is claimed to be prime) and certify to be correct with the ACL2 theorem prover. This means that I must be able to code the test in ...
0
votes
0answers
35 views

For which natural numbers $m,n>1$ does the inequality $2\uparrow^m n>f_m(n)$ hold?

Denote $$f(n,m):=2\uparrow^{m-1} n$$ (See : Wiki ) and $$g(n,m):=f_m(n)$$ (See : Wiki ) It is straightforward to show $f(n,m)<g(n,m)$ for all $m,n>1$ via induction. But for which $m,n>1$ ...
3
votes
0answers
30 views

Is there a name for the least exponent $e$ such that a power of an integer is divisible by another?

Say the primes dividing $m$ also divide $n$. Is there a name for the least exponent $e$ such that $m | n^e$? I can write that explicitly using the prime factorizations of $m$ and $n$, but am ...
2
votes
2answers
28 views

Why is the Apollonian Gasket composed of infinitely many circles?

This famously known fractal has infinitely many circles, however I find it hard to find a rigid proof that confirms how or why this fractal is composed of infinitely many circles (and only circles). ...
2
votes
3answers
55 views

Nonlinear system Diophantus.

In the extant books of Diophantus, are considered in the system of equations. Of interest is the non-linear system of Diophantine equations. Some simple systems from his book manages to solve it. ...
1
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2answers
119 views

a conjectured new generating function of narayana's sequence

In the 14th century ,an Indian mathematician T.V Narayana came up with a sequence now named after him.The sequence satisfies the recurrence $$a_{n}=a_{n-1}+a_{n-3}$$ Starting with $a_{0}=a_{1}=1$, ...
2
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0answers
36 views

Module isomorphisms and coordinates modulo $p^n$

Let $p$ be a prime and $n \in \mathbb{N}$ is such that $p^n > 2$. We let $\alpha \in \mathbb{N}$ be such that $0 < \alpha < n$. Let $R := \mathbb{Z}_{p^n}$ denote the ring of integers modulo ...
0
votes
2answers
71 views

Can this binomial polynomial sum be simplified?

$$\sum_{k=0}^{n} \binom{n}{k} k^d$$ where $d$ is some fixed positive integer. Is this a well known sum that has a faster-than-$O(n)$ evaluation? It looks similar to Faulhaber's formula, except with ...
8
votes
3answers
74 views

Does the sum of reciprocals of primes congruent to $1 \mod{4}$ diverge?

Let $P$ be the set of primes $p$ greater than $3$ such that $p\equiv1 \pmod{4}$. Does the following sum converge or diverge? $$ \sum_{p\in P}\frac{1}{p} $$
3
votes
1answer
86 views

Solving the number theoretic equation $ \sum_{d|n}{d^4}=n^4+n^3+n^2+n+1 $

I found an interesting problem: Find all $n\in\mathbb N$ such that $$ \sum_{d|n}{d^4}=n^4+n^3+n^2+n+1 $$ If we define $s(n)=\sum_{d|n}{d^4}$, we can show, that $s(mn)=s(m)s(n)$ if $\gcd(m,n)=1$. ...
0
votes
1answer
20 views

Condition inverse $p$-adic number

Take $p$ prime, $n \in \mathbb{Z}_{>0}$ and $x \in \mathbb{Z}_p$. Suppose that $p$ isn't a divisor of $$x = (x_j + p^j \mathbb{Z})_{j \in \mathbb{Z}_{>0}},$$ then one can prove that the first ...
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votes
2answers
42 views

Formula to sum up the rational below 1 with bounded demoninator? [closed]

Is there a formula that describes the sum of rationals with a maximum denominator such as 5 that are smaller than one?
7
votes
1answer
42 views

Show that there exist a prime divisor of $\sigma{((2^k)!)}$ which is greater than $2^k$

Let $k$ be a positive integer. Show that there exists a prime divisor of $\sigma{((2^k)!)}$ which is greater than $2^k$, where $\sigma{(n)}$ is the sum-of-divisors function.
0
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0answers
25 views

How to calculate sum of LCMs [duplicate]

How to solve this problem? Given n, calculate the sum LCM(1,n) + LCM(2,n) + .. + LCM(n,n). Is there any way to solve it by math?
9
votes
1answer
60 views

inequality $\max\{a_1,a_2,\cdots,a_n \}\leq {n^2}^{n-1}.$with Egyptian fraction

Let $a_1,a_2,\cdots,a_n $ be positive integer such that$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=1.$ Prove that$$\max\{a_1,a_2,\cdots,a_n \}\leq {n^2}^{n-1}.$$ This Problem from:1
0
votes
1answer
80 views

Show there are infinitely many primes that are equivalent to 1 mod 8.

Show there are infinitely many primes that are equivalent to $1 \pmod{8}$. Hello there! I have been trying to do this problem for a pretty long time with no avail. I noticed that this is really ...
0
votes
4answers
40 views

Proving binomial coefficient formula based on Pascal's triangle

I am trying to practice proving things, and I came across one I wasn't sure about. We already know that $\binom{n}{k}$ is the sum of the two corresponding "parent" entities in Pascal's triangle, ...
4
votes
1answer
34 views

Perfect number in gaussian integers

We have complete description about irreducibles in the ring Z[i],of gaussian integers. Now I was trying to define suitably the notion of "perfect number" in Z[i]. But the problem is unique ...
14
votes
3answers
281 views

Prove that there exists infinitely many positive integers $n$ such that $\sin^2{(na)}+\sin^2{(nb)}\le \frac{2\pi^2}{n}$

Can anyone please help me with the following proof: Prove that there exists infinitely many positive integers, $n$, such that $$\sin^2{(na)}+\sin^2{(nb)}\le \dfrac{2\pi^2}{n}\quad a,b\in \Bbb R$$
3
votes
4answers
129 views

On the near-integer $163/\ln(163)$

This question, concerning the approximation $\frac{163}{\ln(163)}\approx 2^5$, was posted on MO 5 years ago: Why Is 163/ln(163) a Near-Integer?. It was concluded that it had nothing to do with 163 ...
2
votes
2answers
68 views

Prove that every positive integer less than or equal to the square root of a is a near factor of a

In many computer languages, the division operation ignores remainders. Let's denote this by the operation $//$, so for instance $13//3 = 4$. If for some $b$, $a//b = c$ then we say that $c$ is a near ...
0
votes
1answer
36 views

Ring homomorphism from $\mathbb{Z} \to\mathbb{Z}_n$ [closed]

If m ∈ $\mathbb{Z}_n$ and n = 12 for what values of m is the function defined $\phi_m = \mathbb{Z} \to \mathbb{Z}_n, x\to(mx)\mod n$, a ring homomorphism.
4
votes
1answer
42 views

Any correlation to Merten's function?

Here is a plot of partial sums of Liouville Lambda and Moebius Mu: Notice the differences (in green) are tantalizingly close to $-n^{\frac{1}{2}}$. Does this have any correlation to Merten's ...
8
votes
4answers
160 views

Proof of $\sum_{d|n} {\tau}^3(d)=\left(\sum_{d|n}{\tau}(d)\right)^2$ (not standard proof)

I am trying to prove that $\sum_{d|n} {\tau}^3(d)=\left(\sum_{d|n}{\tau}(d)\right)^2$ without using the fact that $\tau$ is multiplicative and products/sums of multiplicative funcions are also ...
1
vote
0answers
25 views

Finding the value of a sum of two Hecke eigenvalues

I did some computations but I am stuck in finding the exression of the sum $$\lambda_f(n^2)+\lambda_f(n)^2 $$ in terms of $\lambda_f(n),$ where $f$ is a modular form for the full modular group. Any ...
3
votes
1answer
39 views

$\binom{n}{k}$ modulo prime power for large $n$ and small $k$

I have to compute several value of $\binom{n}{k}$ mod $p^a$ for prime $p$ over a range of $k$, where $n$ is large and fixed, and $k$ is small and dynamic. Is there a way to speed the process up? If I ...
1
vote
2answers
36 views

Euler's theorem with a product of primes

Could someone tell me why it is, that if you have a product of primes, say 15, then if you use the slightly modified Euler's theorem, then the equation works for every number, not only for relative ...
0
votes
3answers
56 views

Is this number theory proof correct? (irrationality of $\sqrt n$)

I read the following proof in a book while I seeing the proof that is irrational if $n$ is not a perfect square number. The proof is as follows- Let, if possible, there exists rational number ...
8
votes
5answers
115 views

$32$ Goldbach Variations - Papers presenting a single gem in number theory or combinatorics from different point of view

A short time ago I found the nice paper Thirty-two Goldbach Variations written by J.M. Borwein and D.M. Bradley. It presents $32$ different proofs of the Euler sum identity \begin{align*} ...
19
votes
2answers
170 views

For all $n$ there exists $x$ such that $\varphi(x)<\varphi(x+1)<\ldots<\varphi(x+n)$

Let $\varphi$ be the Euler's function, i.e. $\varphi(n)$ stands for the number of integers $m \in \{1,\ldots,n\}$ such that $\text{gcd}(m,n)=1$. Let $n\ge 2$ be a positive integer. Show that there ...
4
votes
2answers
199 views

Is the limit $ \lim_{n\to \infty}\left(\sum^{n}_{r=0} \binom{n}{r}\big/{n^{r}(r+3)}\right)$ rational or irrational?

How can I prove that the result of the following limit is rational/irrational?$$ \lim_{n\to \infty}\left(\sum^{n}_{r=0} \frac{\binom{n}{r}}{n^{r}(r+3)}\right)$$ Would solving this limit satisfy? How ...