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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2
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1answer
176 views

How to calculate this sum like Gauss sum.

I would like to calculate the following sum, which looks like a Gauss sum. Let $n$ be a natural number and let $a,b$ be integers. Denote by $e(x)=e^{2\pi i x/n}$. Consider the sum $$ \sum_{1 \leq j, ...
0
votes
0answers
61 views

Solve $x^2 + 2 = y^3$ for integer $x$ and $y$ [duplicate]

I am asked to find all integers $x$ and $y$ which satisfy $x^2 + 2 = y^3$. I am given the hint that I should work in the unique factorization ring $\mathbb{Z}[\sqrt{-2}]$. So I could write the ...
3
votes
3answers
89 views

find a number such that, for all $a$ in $\{0,…,1926\}$, $a^x \equiv a \mod 1926$.

I don't want the answer, but I need some help on how to figure out the answer. If you could point me in the direction of a useful math theorem or technique it would much much appreciated. Also, I am ...
0
votes
2answers
33 views

On the Diophantine Equation $(x-h)^2+(y-k)^2=c$

I am just curious about the equation of the circle centered at (h,k) whose form is we know $(x-h)^2+(y-k)^2=r^2$. If we consider its solution over the set of integers then we have a Diophantine ...
12
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3answers
99 views

Prove that $2$, $3$, $1+ \sqrt{-5}$, and $1-\sqrt{-5}$ are irreducible in $\mathbb{Z}[\sqrt{-5}]$.

So the Norm for an element $\alpha = a + b\sqrt{-5}$ in $\mathbb{Z}[\sqrt{-5}]$ is defined as $N(\alpha) = a^2 + 5b^2$ and so i argue by contradiction assume there exists $\alpha$ such that $N(\alpha) ...
2
votes
1answer
34 views

Embeddings of a subfield of $ \mathbb{C} $

I'm trying to understand / solve the following problem: Let $ L \subset \mathbb{C} $ be a field and $ L \subset L_1 $ its finite extension ($ [L_1 : L] = m $). Prove that there are exactly $ m $ ...
1
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2answers
23 views

How to prove that N(u) = 1 if and only if u is a unit in $Z[\sqrt-5]$

The norm of an element $u=a+b\sqrt-5$ in $Z[\sqrt-5]$ is defined as $N(u)= a^2 +5b^2$, now if $N(u) = 1$ then $a^2+5b^2 = 1$ but then how would i prove that it's a unit !?
1
vote
1answer
100 views

Why don't we use Presburger's arithmetic instead of Peano's arithmetic?

I was reading about quantifier elimination and discovered the Presburger Arithmetic, the article mentions two points about it: It is decidable, complete and consistent. It omits multiplication ...
2
votes
0answers
85 views

The curse of Collatz [closed]

With this playful title I wanted to open a discussion on a personal attempt to prove this conjecture after several sheets and energy wasted (I'm not English, sorry if my language isn't perfect). The ...
1
vote
2answers
60 views

Which arrangement produces the largest number?

I learnt that the power tower $2\uparrow3\uparrow4\uparrow...\uparrow n$ is larger than any power tower with a different order of the numbers $2,3,4,...,n$. Is this also true for conway-chains and ...
3
votes
1answer
82 views

Solve the eqation $a^3+2b^3+4c^3=6abc+1$

Find all integers $a,b,c>2010$ so that $a^3+2b^3+4c^3=6abc+1$. If there are no solutions then prove it. As for now I only tried to use the identity ...
1
vote
1answer
67 views

How to evalute the $\sum_{n=1}^\infty \dfrac{\phi(n)}{2^n -1}$?

In the above question $\phi$ is Euler's phi-function. This problem belongs to IMO shortlist. All my efforts doesn't lead to any good result.
2
votes
4answers
68 views

Can one generate a sequence of natural numbers whose density has a given distribution?

Suppose $\{ p_{k} \}$ is a collection of real numbers with the following properties: 1) $p_k \in (0,1)$ $~~~~$(i.e. $0$ and $1$ are not allowed values) 2) $\sum_{k=1}^{\infty} p_k =1$ An ...
7
votes
3answers
69 views

Find all $n$ so that $s(2^n)=s(5^n)$

Let $s(k)$ be the sum of digits of the number $k$. Find all $n$ so that $s(2^n)=s(5^n)$ I have got that $n=3$ is one solution. And looking modulo 9 it's easy to get that 3 has to divide $n$.
0
votes
1answer
24 views

Norms on a Euclidean domain

A norm $N$ on a Euclidean Domain $R$ is a function $N:R \longrightarrow \{0,1,2,...\}$ such that (i) $N(a) = 0 \longleftrightarrow a = 0$ (ii)$N(ab) = N(a)N(b)$ (iii) if $b \neq 0$, then $a=qb + ...
0
votes
2answers
44 views

How would one prove this flavour of the triangle inequality?

I have to prove $|z_1 - z_2| \leq |z_1|+|z_2|$ where $z_1,z_2$ are in $\mathbb{C}$. What I wrote down is: $$|z_1| = |z_1+z_2-z_2| \geq |z_1-z_2|-|z_2|\implies |z_1|+|z_2|\geq |z_1-z_2|,$$as desired. ...
2
votes
0answers
59 views

How to solve this equation $x^5 +4^y=2013^z$ in positive integers?

I think to solve the equation in positive integers. It appears in a contest and I don't remember where. I obtain that $x$ must be an odd number and further $x=1 \, mod\, 4$. Any hint is appreciated.
1
vote
1answer
23 views

Euclids Division Algorithm problem.

Show that if $a$ and $b$ are positive integers, then there are unique integers $q,r$ and $e = \pm 1$ such that$ a = bq + er $, where $-b/2 < r \leq b/2$. I am always having trouble with these. :( ...
1
vote
2answers
64 views

System of congruences and Chinese remainder theorem

Find all the integers satisfying this system of congruences $$\begin{cases} x \equiv 2 \pmod 5\\ x \equiv 1 \pmod {10}\\ x \equiv 0 \pmod 3 \end{cases} $$ I think you use Chinese remainder theorem ...
1
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0answers
17 views

Independence of FLT over weak systems

It is known that Fermat's last theorem can be proven in finite-order arithmetic (e.g. accoridng to this site). This is still an extremely high upper bound on proof complexity (for example, compared to ...
0
votes
1answer
20 views

Permutation: How many numbers of n digits are possible for which product of its digits is a perfect square.

I need to find total numbers from 1 to 10000 whose product of digit is a perfect square. eg: 49 (4*9=36), 236 ( 2*3*6=36) etc. Till now i have figured out these things: 1) For a number to be a ...
0
votes
1answer
34 views

Is there a relation between product of digits of a number and perfect square?

I want to find all numbers less than N whose product of digits is a perfect square. for example if N is equal to 100 then some of possible numbers are 22 (2*2), 49 (4*9=36), 2*8, 8*2 etc. I was ...
3
votes
0answers
60 views

Questions about decimal expansion being able to represent all real numbers

I read this in several books, and there's a Wikipedia article unquestionably stating that reals must be representable by means of regular language generated from finite alphabet. My questions are: ...
3
votes
2answers
116 views

Can anyone sketch the proof or provide a link that there is always a prime between $n^3$ and $(n+1)^3$

In a recent forum discussion on number theory, it was mentioned that A. E. Ingham had proven that there is always a prime between $n^3$ and $(n+1)^3$. Does anyone know if there is a link available on ...
3
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3answers
51 views

Mathematical induction involving inequalities and congruences

I have the following two problems: "Prove each of the following statements by induction for all positive integers $n$:" $2\cdot7^n \equiv 2^n\cdot(2+5n) \bmod 25 \quad$ <-- I have been going at ...
9
votes
3answers
76 views

Is there a monotonic $f$ such that $\sum f(n)$ diverges but $\sum f(p)$ converges?

(where the former summation is over natural numbers $n$ and the latter is over prime numbers $p$, and $f: \mathbb{N} \to \mathbb{R}$ is a monotonic function.) For the class of functions $f_s(n) = ...
1
vote
1answer
28 views

algebraic conjugate and sum of roots of unity

In above lemma, why $|a'| \leq 1$ still holds? I didn't see how it relates to "algebraic conjugate of a root of unity is also a root of unity", since $a$ is the sum of unity. (definition of ...
2
votes
1answer
97 views

Solutions to $y^2 = x^3 + k$?

As you know, the equation $y^2 = x^3 + k$ for $k = (4n-1)^3 - 4m^2$, with $m, n \in \mathbb{N}$ and no prime number that p is congruent to 1 modulo 4 count m, don't have any answer and its proof can ...
12
votes
1answer
119 views

Are there infinitely many pairs of primes where one divides one more than the square of the other?

I have the following question on number theory that is eating my head. Are there infinitely many primes $p,q$ such that $p | (q^2 + 1)$ and $q | (p^2 + 1)$? I can see $13,5$ and $2,5$ has the ...
3
votes
0answers
108 views

What is the value of $\sum_{p\le x} 1/p^2$?

My question is, what is the value of $$\sum_{p\le x} \frac{1}{p^2}?$$ More generally, what is the value of $$\sum_{p\le x} \frac{1}{p^n}?$$ How can we find it? For $\sum_{p\le x} 1/p$ the idea was ...
0
votes
0answers
17 views

Is it decidable whether there are finitely or infinitely many positive integers n such that the (2^n)+1th last digit of 3^(2^n) in base 2 is 1.

It seems so obvious that there are infinitely many such n because using a probabilistic arguement, it seems like there's no chance of there only being finitely many of them. Yet, it seems impossible ...
11
votes
1answer
151 views

Is $\sum\frac1{p^{1+ 1/p}}$ divergent?

Is $\displaystyle\sum\frac1{p^{1+ 1/p}}$ divergent? How can we prove that it is divergent or convergent in analytic number theory? I know what bound of the n-th prime number is, and that its order is ...
1
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0answers
42 views

If $m^x$ an integer for all naturals $m$ then $x$ and integer [duplicate]

I have seen this result somewhere: If $x$ is a positive real number such that $m^x$ is an integer for all natural numbers $m$, then $x$ is an integer. It is quite interesting and I have some ...
10
votes
2answers
209 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
119 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 ...
2
votes
2answers
47 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
88 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 = ...
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0answers
40 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
43 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
46 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
66 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 ...
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0answers
20 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
517 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 : ...
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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
44 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
47 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
115 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 = ...