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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10
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2answers
201 views

Gardner riddle on mathemagicians

A cute riddle (but maybe not so easy!) from Gardner: At a gathering of mathemagicians, the Grand Master and his 8 disciples are seated at a round table. The Grand Master will judge each of his ...
3
votes
0answers
112 views

Some Diophantine problems for equal sums with high powers

Given rationals $R = a,b,c,d,e,f$. Define, $$F_n = a^n+b^n+c^n-(d^n+e^n+f^n)\tag1$$ Finding certain equal sums of like powers that are multi-grades for high powers lead to two questions: If ...
1
vote
2answers
38 views

Where have I gone wrong in calculating norm residue symbol.

It is well known that the reciprocity map in local class field theory gives reciprocity map in Global class field theory. Namely, if $(\cdot,L_\beta/K_P):K_P^\times \rightarrow G(L_\beta/K_P)$ is the ...
1
vote
1answer
47 views

To find all positive integers $n$ such that $n|a^2-1 ; a \in \mathbb Z \implies n|a-1 $ , or $n|a+1$

How do we find all positive integers $n$ such that $n|a^2-1 ; a \in \mathbb Z \implies n|a-1 $ , or $n|a+1$ ? I have found that for any odd prime $p$ and $n \in \mathbb Z^+$ , $p^n|a^2-1 ; a \in ...
1
vote
4answers
87 views

Could someone be so kind as to explain this little summation to me?

So basically, the wording in this question, to me, is weird. It goes as follows: Explain why the following formula gives the power $e$ of a given prime $p$ in $n!$: $$e = ...
1
vote
0answers
39 views

An elliptic curve for the multigrade $\sum^8 a_n^k = \sum^8 b_n^k$ for $k=1,2,3,4,5,9$?

I. The first solution to, $$\sum^6_{n=1} a_n^9 =\sum^6_{n=1} b_n^9$$ was found in 1967 by Lander et al. In 2010, Bremner and Delorme found it had the highly structured form, $$\small(u + 9)^k + (u ...
1
vote
0answers
42 views

Repeating decimal algorithm

I was working on a problem where I needed to prove things in base 10, like: "11 divides $a$ if and only if 11 divides $a_0-a_1+a_2-\cdots$" where ...
1
vote
1answer
44 views

Why does it suffice to show it for positive integers?

I am looking at the proof of the product formula theorem: For each $x \in \mathbb{Q}$, it holds $$\prod_{p \leq \infty} |x|_p=1$$ The proof starts by this: It is enough to show it for ...
0
votes
3answers
31 views

Write $M^+=\{n \in M: n>0\}.$ Is $M^+$ non-empty? Explain.

Let $a$ and $b$ be two positive integers and $M$ the set of all integer linear combinations of $a$ and $b$. Write $M^+=\{n \in M: n>0\}.$ Is $M^+$ non-empty? Explain. Just to provide more detail ...
4
votes
1answer
43 views

Is there a more precise modified stirling's approximation formula for calculating n!?

I am trying to solve a problem of competitive programming Consider two integer sequences $f(n) = n!$ and $g(n) = a^n$, where $n$ is a positive integer. For any integer $a > 1$ the second ...
1
vote
1answer
57 views

Fundamental Unit In Algebraic Fields

Say we have an algebraic field with an infinite amount of units. If one multiplies two units one obtains another unit. In some cases, all units are powers of just one unit ( that's the fundamental ...
1
vote
0answers
19 views

Does Idele group of norm 1 preserved by the norm?

I should explain my question in detail as of now I'm sure it makes no sense. Let $K$ be a global field (in particular I care about the characteristic $p$ case.) Then its Idele group $I_K$ has a ...
35
votes
1answer
473 views

Numbers $n$ such that the digit sums of $n, n^2,\cdots,n^k$ coincide.

Let $S(n)$ be the digit sum of $n\in\mathbb N$ in the decimal system. When I was playing with numbers, I noticed the followings : ...
1
vote
0answers
56 views

FLT (Fermat): Combinatorial approaches?

Such a simple equation like $x^n+y^n=z^n$ is bound to have a nice/natural combinatorial interpretation. One very crude one is: Let the number of ways of choosing $n$ objects from $x$ objective, ...
0
votes
1answer
42 views

On infinitude of primes of certain form. [duplicate]

We know that there are infinite number of primes so as there are infinite number of primes of the form $4n+3$ where $n\in Z^+$. A note on Burton's book (Elementary Number Theory) somehow says that ...
4
votes
1answer
41 views

Why is Minkowski's Theorem so powerful?

Minkowksi's Convex Body Theorem is evidently pretty powerful, as it yields swift proofs of Fermat's Two Square and Lagrange's Four Square Theorems. Also, Minkowski's bound on class number and the ...
2
votes
4answers
112 views

Rationality of $e + \pi$

I found just one question similar to this, but it had been edited, so hopefully this isn't asked too often. Given the formulas via infinite sums for expressing $e$ and $\pi$... $$ e = ...
0
votes
2answers
40 views

Concatenation of integers

Background Direct addition is what some call 'hobo math' (no insult intended); for example, by direct addition, $2+2=22$, and $13+36 = 1336$. I know the process for writing out a number given its ...
0
votes
1answer
20 views

CHECK: Let $M$ be the set of all integer linear combinations of $a,b \in \mathbb{Z}$. Prove that $M$ is closed under addition and multiplication

Let $M$ be the set of all integer linear combinations of $a,b \in \mathbb{Z}$. Prove that $M$ is closed under addition: $u,v \in M$ implies $u+v \in M$ $M$ is closed under multiplication by any $k ...
2
votes
1answer
29 views

How can one show that $\prod_{n<p\leq2n}p\leq C(2n,n)$?

I am trying to rove that $\prod_{n<p\leq2n}p \leq C(2n,n) \leq 2^{2n}$, where $C(2n,n)= \frac{2n!}{n! n!}$ and $p$ is prime. I can prove the second part by induction, but first part induction ...
2
votes
1answer
54 views

How to find the restricted partition of n into k *distincts* parts between a finite set [1;r]?

It seems to be an opened question. Indeed, it is easy to find: the number of partitions of n into k distinct parts the number of partitions of n into k parts the number of partitions of n into k ...
10
votes
1answer
215 views

First Order Logic: Prove that the infinitely many twin primes conjecture is equivalent to existence of infinite primes

I'm learning First Order Logic independently using a college textbook. I've been doing some self exercise question in it and came across this one, which I can't seem to figure out how to do: Let ...
2
votes
3answers
44 views

Calculating the nth root in simple calculator?

Recently during my physics class that how to take a cube root in a simple calculator. Follow the steps given below Step1. Press the square root button 10 times Step2. subtract 1 from it Step3. ...
4
votes
1answer
34 views

determinant of divisor functions

Let A be a $(n-1) \times (n-1)$ matrix whose entries $a_{ij}=d(\gcd(i+1,j+1))$. $d(n)$is the number of divisors of $n$. It seems that the determinant of it is the number of square-frees less than or ...
2
votes
0answers
47 views

Integration over ideles over $\Bbb{Q}$ , Tate`s thesis special case

Let $f\in S(A_\Bbb{Q})$ that is $f$ is adelic Schwartz-Bruhat function over $\Bbb{Q}$, such that all its components in the finite places are characteristic functions of the corresponding ring of ...
6
votes
1answer
217 views

If such $1+4\sin{10^\circ}=a+b\sin{c^\circ}$ How find this integer $a,b$

Interesting problem: Assmue that: $$1+4\sin{10^\circ}=a+b\sin{c^\circ}$$ where $a>0$,and $a,b,c$ are integers and $0<c<90^\circ$, show that $$a=1,b= 4,c=10$$ is unique solution ...
2
votes
0answers
30 views

an exponential sum involving quadratics

Let $q$ be a prime and $a,b\in \mathbb{Z}$. Can we compute $$\sum_{k=1}^{q}e^{2\pi i\frac{ak^2+bk}{q}}$$ in terms of $a,b,q$? The sum is easy when $a=0$, but what about the case $a\neq 0$?
1
vote
2answers
24 views

Show that two different representations to the base $k$ represent two different integers

I would like to show: Given two distinct, positive, integer representations in base $k$, say $\sum_{i=0}^na_ik^i$ and $\sum_{i=0}^mb_ik^i$ where $a_n \neq 0 \neq b_m$ and $a_i,b_i \in \{0,1,\ldots ...
2
votes
3answers
49 views

CHECK: Show that if $b$ is an odd number, then $gcd(2a,b)=gcd(a,b)$

Show that if $b$ is an odd number, then $$gcd(2a,b)=gcd(a,b)$$ $\textbf{Proof:}$ We will prove that $(a,b)$ and $(2a,b)$ have the same set of divisors. Assume that $b$ is an odd number. ...
5
votes
1answer
82 views

Question about “baffling” umbral calculus result

I am reading a paper here and I've come to a particular passage that is confusing me. It comes on page 2 of the attached paper and it deals with the binomial theorem... The passage lays the ...
2
votes
0answers
46 views

Tate thesis : Global functional equation [closed]

It will be very helpful if someone tells me how to do EXERCISE 1 here. I have done part $2$. I cannot do part $1$ and part $3$. I tried part $1$ by decomposing $Z(f,s)$ as the product of its local ...
1
vote
1answer
27 views

What are sharp lower and upper bounds of the fast growing hierarachy?

With fast growing hierarchy, I mean the Wainer hierarchy, which starts with $$f_0(n)=n+1$$ $$f_1(n)=2n$$ $$f_2(n)=2^n n$$ A lower bound for $f_m(n)$ is $2 \uparrow^{m-1} n$. If $f(n,m):=2 ...
0
votes
3answers
75 views

find all solutions of $(x^2+y^2+z^2-1)^2+(x+y+z-3)^2=0$

find all solutions of $(x^2+y^2+z^2-1)^2+(x+y+z-3)^2=0$ I tried to expand this but its yielding nothing and becoming clumsy.How to do this?
1
vote
1answer
44 views

How to solve $x^2-4y=m^2$ where $m$ is given?

Respected all Kindly help me to solve the following diophantine equation. The equation is given by $x^2-4y=m^2$ where $m\in \mathbb Z$ is given. How to solve this equation in integers? I have read ...
0
votes
1answer
45 views

How to solve $(xy)^2+a(xy)+bx+cy+d=0$ in integers?

Respected all. We know that $x^2+y^2+2gx+2fy+c=0$ represents a circle and the parametric solution for it is $x=\cos t, y=\sin t$. But I was wondering what would happened for the following equation ...
34
votes
1answer
1k views

Why does this test for Fibonacci work?

In order to test if a number $A$ is Fibonacci, all we need to do is compute $5A^2 + 4$ and $5A^2 -4$. If either of them is a perfect square, the number is Fibonacci, otherwise not. Why does this test ...
0
votes
0answers
33 views

Does this notation mean what I intend?

I was looking at divisibility rules earlier today and noticed that several of them had the same form, i.e. truncating the last digit and then adding or subtracting a multiple of it to the truncation. ...
1
vote
1answer
35 views

different prime factor

Let $k$ be an positive integer and $m$ is the odd positive integer. Prove that there exist a natural number $n$ such that $ m^n + n^m $ have at least $k$ different prime factor
2
votes
0answers
64 views

Pythagorean triple problem

I am doing research on perfect cuboids, and I'm looking for values $a,b,c$ such that the following is integer, and I'm not sure how to continue this. Any suggestions are appreciated! $PED$ is a very ...
1
vote
0answers
64 views

if $\forall n; n^{\alpha} \in \mathbb{N}$ then $\alpha \in \mathbb{N}$ [closed]

Let $ \alpha \in \mathbb{R} $ such that: $\forall n \in \mathbb{N}; n^{\alpha} \in \mathbb{N} $ prove that $\alpha \in \mathbb{N}$
1
vote
2answers
53 views

Diophantine Equation With Varying Exponents

I am considering the following Diophantine Equation - the approach I tried became the study of too many different cases - so many that I left it and tried to find an easier way. I wonder if anyone ...
1
vote
1answer
54 views

What is the number of elements in the set {(x,y): x|n, y|n, gcd(x,y)=1} ?.

Sloane's OEIS A048691 gives an explicit formula:(2*e(1)+1)*(2*e(2)+1)***(2*e(r)+1) where the e(i)'s are the exponents in the prime factorization of n. It turns out that the same formula counts the ...
3
votes
1answer
74 views

How to prove that $2^x,3^x,5^x\in\mathbb N$ implies $x\in\mathbb N$? [duplicate]

Let $x\in\mathbb R$ and suppose that $2^x,3^x$ and $5^x$ are all integers. Does it imply that $x$ is also necessarily an integer? I read somewhere that the answer is "Yes" and a proof is known, but I ...
2
votes
1answer
34 views

twisted gaussian integers; complex plane with a different basis

I'm trying to understand a kind of twisted form of Gaussian integers. They are defined via $$ w = e^{i \frac{2}{3} \pi}\\ R = \{ m + nw \mid m,n \in \mathbb{Z} \}$$ I tried to picture them by using ...
4
votes
2answers
84 views

How find a closed form for the numbers which are relatively prime to $10$,

Interesting Question Let $a_n$ be the positive integers (in order) which are relatively prime to $10$. Find a closed form for $a_n$. I know ...
1
vote
1answer
54 views

Are there solutions to FLT which are linearly independent over $\mathbb{Z}$

Specifically, I would like to know if there is some $R$, where $R$ is a ring with unity $\mathbb{Z} \subseteq R$ there are $x,y,z \in R$ and a prime $p \in \mathbb{Z}$ such that $x^p + y^p + z^p = ...
13
votes
1answer
138 views

Numbers that are the sum of the squares of their prime factors

A number which is equal to the sum of the squares of its prime factors with multiplicity: $16=2^2+2^2+2^2+2^2$ $27=3^2+3^2+3^2$ Are these the only two such numbers to exist? There has to be an ...
1
vote
1answer
26 views

Show that the idele group of a number field is locally compact

Let $k$ be a number field and $M_k$ the canonical set of places of $k$. Also let $S_\infty$ be the set of Archmedean places of $k$. For each $v\in M_K$ let $k_v$ be the completion of $k$ wrt an ...
1
vote
1answer
53 views

Is there a solution of $2^n+3^m-5^k=1$

I dont know if there is a solution of $$2^n+3^m-5^k=1$$ especialy if the values of $n$, $m$ and $k$ integer numbers greater than $100$
-1
votes
0answers
46 views

Show that the product is a perfect square [duplicate]

$n$ is a positive integer. Let $P_k(n)$ be the product of all positive divisors of $n$ that are divisible by $k$(empty product is equal to $1$). Show that $P_1(n)P_2(n)...P_n(n)$ is a perfect square.