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
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0answers
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2
votes
1answer
52 views

Showing there is no triplet of positive integers $(a,b,c)$ satisfying $a^7+b^7=7^c$

Show that $$a^7+b^7=7^c\tag{$\star$}$$ has no positive integer solutions $(a,b,c)$. I tried a general and way too long approach, which I had posted as an answer, but it turned out to contain a ...
2
votes
2answers
50 views

Is there always a square between two consecutive cubes?

Is there always a square between two consecutive cubes? I just thought of this question. It seems really simple and the answer is probably yes. Edit: I should have given this more than 2 seconds of ...
-1
votes
2answers
70 views

Find the smallest natural number $n$

Find the smallest natural number $n$ such that rightmost digit is $6$ and when we deleted that digit $6$ and add it to the left of the number we get $4n$. Example of the operation: $123456$ becomes ...
5
votes
2answers
54 views

$L$-function, easiest way to see the following sum?

What is the easiest way to see that$$\sum_{(m, n) \in \mathbb{Z}^2 \setminus \{0, 0\}} (m^2 + n^2)^{-s} = 4\zeta(s)L(s, \chi)?$$Here $\chi$ is the homomorphism $(\mathbb{Z}/4\mathbb{Z})^\times \to ...
0
votes
0answers
10 views

Extending 2-adic valuation to real numbers

When proving Monsky's theorem, one of the steps, which, from what I have so far seen, no proof can avoid, is extending the 2-adic valuation to all real numbers, so that it still satisfies ...
0
votes
1answer
52 views

Expansion of factorial of a natural number as a summation

Factorial of any natural number $n$ (i.e. $n\in N$) can be expanded as a summation $$n!=1+\sum_{i=1}^{n-1}(i\times i!)$$$$=1+1\times 1!+2\times 2!+3\times 3!+4\times 4!+..............+(n-1)\times ...
2
votes
1answer
38 views

Fermat's Theorem and primitive $n$th roots of unity

Fermat's little theorem states that if $p$ is a prime number, then for any integer $a$ such that $p \not \mid a$ we have that $a^{p − 1} \equiv 1 \operatorname{mod} p$. Suppose $p =17$, then we know ...
0
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0answers
26 views

How do I calculate these sum-of-sum expressions in terms of the generalized harmonic number?

I know that $$\sum_{m=2}^k\sum_{n=1}^{m-1}(nm)^{-s}=\frac 12((H_k^s)^2-H_k^{(2s)})$$ and $H_k^s=\sum_{n=1}^kn^{-s}$ But, how would I go about finding identities in terms of the harmonic number like ...
4
votes
4answers
69 views

what is the greatest integer that divides $p^4-1$ for every prime number p greater than 5

what is the greatest integer that divides $p^4-1$ for every prime number p greater than 5(this is a gre subject math problem) I think that $p^4-1=(p^2+1)(p-1)(p+1)$,so 8 must divide all the $p^4-1$ ...
5
votes
1answer
55 views

Find all primes $a,b,c$ and integer $k$ satisfying the equation $a^2 + b^2 + 16 c^2 = 9k^2 +1$

This was a problem in this year's Junior Balkan Olympiad. So here's what I did first: If $a,b,c,k$ satisfy the conditions, then they satisfy the congruence: $$a^2 +b^2 + c^2 \equiv 1\pmod 3$$ ...
1
vote
1answer
39 views

If Wieferich primes are finite…Then what?

I am wondering if $1093$ and $3511$ are the only Wieferich Primes, then what would it imply? (A wieferich prime is a prime satisfying the congruence $2^{p-1}\equiv 1\ mod \ p^2 $). I know of 3 cases; ...
1
vote
1answer
26 views

Identity for $L(s,\chi)L(s,\bar\chi)$

I was told recently that there is an identity roughly of the form $$L(s,\chi)L(s,\bar\chi)=\zeta(s)^2$$ If true, it seems like there should be a short proof of this. Could someone supply a ...
0
votes
0answers
15 views

Rational Right Triangle Problem & Elliptic Curves

If we know the sides of a right triangle with rational sides and area=7, from this triangle, how can we get the right triangle with rational sides and area=14? Or the question can be the other way ...
1
vote
1answer
21 views

Remainder regarding some identity regarding primes.

How do I show the following identity: $$\sum_{p\ is \ prime} \log (1-1/p^s)=\sum_{n=2}^\infty (\pi(n)-\pi(n-1))\log(1-1/n^s)$$ A hint is best. Thanks.
6
votes
1answer
32 views

Quadratic field, $O_K/\mathfrak{p} = \mathbb{F}_p$, $O_K/pO_K$ is a finite field of order $p^2$.

Let $K$ be a quadratic field $\mathbb{Q}(\sqrt{m})$ where $m$ is a square free integer, and let $p$ be a prime number which does not divide $2m$. Where can I find a reference to a proof of the ...
0
votes
0answers
25 views

Dirichlet characters with values in a finite field

Although the classical Dirichlet characters are complex valued, it seems to me rather useful that the characters attain values in a finite field; thus homomorphisms from $\mathbb{Z}_N^*$ to ...
1
vote
1answer
58 views

Infinitude of the primes $p\equiv1 \operatorname{mod} n$

$\textbf{Theorem:}$ Fix $1 < n \in \mathbb{Z}$. There are infinitely many primes $p\equiv1 \operatorname{mod} n$. $\textbf{Proof}$ Recall that the $n$-th cyclotomic polynomial $\Phi_n(x)$ is ...
0
votes
2answers
37 views

Can you express the fraction 1/0 using imaginary units in any way possible? [on hold]

Although in basic textbooks, 1/0 is undefined or something along those lines, can it be expressed using complex numbers (i)? One way I propose is to use a function such as 1/x and use that function in ...
7
votes
1answer
82 views

What is the Galois group of $\mathbb{Q}_p[\zeta] / \mathbb{Q}_p$, where $\zeta$ is a $p^r$th root of unity?

Let $\zeta$ be a $p^r$-th root of unity and $\mathbb{Q}_p$ the p-adic numbers. I would like to know the Galois automorphisms of $\mathbb{Q}_p[\zeta] / \mathbb{Q}_p$. I already know, the degree of ...
0
votes
0answers
40 views

$\eta(1) = \ln(2)$ proof using Abel's Theorem

Hi I was just wondering how does one justify $\eta(1) = \ln(2)$. Looking at the power series for $\ln(1+x)$ we have \begin{equation} \ln(1+x)= \sum_{n=1}^{\infty} \frac{(-1)^{n+1}x^{n}}{n} ...
0
votes
1answer
26 views

Elementary number theory proofs using functions

The functions $f$ and $g$ are defined by $f(x) =$ remainder when $x^2$ is divided by $7$. $g(x) =$ remainder when $x^2$ is divided by $5$. (a) Show that $f(5)=g(3)$ (b) If $n$ is an integer, ...
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0answers
40 views

Does the Collatz conjecture imply the Well-Ordering Principle?

It seems to me that it does, even trivially so. My reasoning is as follows: Suppose that H is a nonempty set of positive integers. Case 1. 1 is a member of H. Then H obviously has a smallest member, ...
3
votes
1answer
121 views

Prove result without using Fermat's Last Theorem

Is there a way to prove that for $x,z,n \in \mathbb{Z}$, $x > 0$, $z > 0$, $n > 2$, the equation $$ \sum_{k = 0}^{n - 1}{\binom{n}{k} x ^k} = z ^ n $$ has no solution, without using ...
6
votes
1answer
66 views

Is $\pi(n)$ a Rational Function?

Are there some two-variable polynomials $P(n,\log n)$ and $Q(n,\log n)$ which we have the bellow equation for prime counting function $\pi(n)$ for $n \in \mathbb{n}$? $$\pi(n) = \Bigl{\lfloor} ...
1
vote
1answer
41 views

If $a|(p+1)$ for all but finitely many $p=3 (\text{ mod } 4)$ then $a$ divides $4$

I have the following question: Let $a$ be an integer such that $a$ divides $p+1$ for all but finitely many primes $p=3 \text{ mod } 4$ Can we conclude that $a$ must divide $4$? How we can prove ...
2
votes
3answers
75 views

Sum of super exponentiation

$f(x,n)=x^{2^{1}}+x^{2^{2}}+x^{2^{3}}+...+x^{2^{n}}$ Example: $f(2,10)$ mod $1000000007$ = $180974681$ Calculate $\sum_{x=2}^{10^{7}} f(x,10^{18})$ mod $1000000007$. We know that $a^{b^{c}}$ mod ...
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0answers
35 views

what is the sum of these 1230 sequences such as 1,2,1,2,2,1,2,2,2,1,2,2,2,2… A. 2411 B. 2412 C. 2413 D. 2414 [on hold]

what is the sum of these 1230 sequences such as 1,2,1,2,2,1,2,2,2,1,2,2,2,2..... A. 2411 B. 2412 C. 2413 D. 2414
2
votes
3answers
104 views

Binomial Coefficient Computation by Dividing Consecutive Terms

If I take the binomial coefficient: $$\frac{n!}{k! (n-k)!}$$ and I want to know the result of 10 choose 4 and I proceed to do the computation $$\frac{7*8*9*10}{1*2*3*4}$$ by first dividing and then ...
3
votes
1answer
41 views

Prime Zeta Function proof help: Why are these expressions not equal?

I was trying to create a formula for the Prime Zeta function and I partially succeeded except for one frustrating error. I was only able to formulate an approximation. Consider the following sum: ...
1
vote
1answer
45 views

How does the Riemann summation connect with prime numbers [on hold]

The Riemann sum goes as the sum of $\dfrac{1}{1^s}+\dfrac{1}{2^s}+\dfrac{1}{3^s}+\ldots$ It has been said that if $\frac{1}{2}+ix=0$ for $s$ ($x$ can be any number), then the sum will have massive ...
0
votes
0answers
27 views

Why does the Harmonic series diverge? [duplicate]

I am aware of Oresme's proof of its divergence(http://mathworld.wolfram.com/HarmonicSeries.html), but this proof could be applied to the sum of all natural numbers and it would still be valid. Yet, ...
4
votes
0answers
40 views

Substitutions that transform Fermat Equations to Elliptic Curves

I was reading Chapter 1 of Elliptic Curves - Number Theory and Cryptography by Lawrence C Washington. He was considering Fermat equations $$a^4+b^4=c^4\text{ and }a^3+b^3=c^3.$$ For the 1st equation, ...
2
votes
2answers
52 views

Number Theory- Mathematical Proof

I am trying to prove a theorem in my textbook using another theorem. What I need to show that if a,b, and c are positive integers, show that the least positive integer linear combination of a and b ...
1
vote
0answers
30 views

Notation for indexing the factorizations of a number?

Background Given any $n \in \mathbb{N}$, the ordered factorization count of $n$ can be computed and is traditionally written $H(n)$. This is, essentially, the number of unique decompositions of $n$ ...
2
votes
1answer
63 views

prove that this number contains two equal digits

We delete the first digit from the number $7^{1996}$ and then we add it to the remaining number, repeat this until we get a number consisting of $10$ digits, prove that, this number contains two equal ...
0
votes
1answer
15 views

Definition of “Contractive Invariant Plane”

Can someone please explain the definition of a contractive invarient Plane found in: the paper It is nearly at the very beginning of the Introduction. By contractive do they mean a contractive map? ...
6
votes
1answer
44 views

Cyclotomic polynomials, properties.

Let $F$ be a field of characteristic prime to $n$, and let $F^a$ be an algebraic closure of $F$. Let $\zeta$ be a primitive $n$th root of unity in $F^a$. I know that the monic polynomial $\Phi_n(X)$ ...
0
votes
0answers
7 views

Comparing conway chains

See https://en.wikipedia.org/wiki/Conway_chained_arrow_notation for the details how conway chained arrow notation works. I want to calculate the approximate value $n$ such that $$n\rightarrow ...
3
votes
1answer
93 views

To find positive integers $n$ such that $\dfrac {n(n+1)(n+2)}6$ is a perfect square [duplicate]

How many positive integers $n$ are there such that $\dfrac {n(n+1)(n+2)}6$ is a perfect square ? I know $n=1 , 2$ works ; are there any more ? Are there only finitely many such $n$ ?
6
votes
1answer
132 views

To solve $n(n+1)(n+2)=6m^3$ in positive integers $m,n$

How to find all positive integers $m,n$ such that $n(n+1)(n+2)=6m^3$ ? I can see that $m=n=1$ is a solution , but is it the only solution ?
2
votes
0answers
15 views

Show that the equation has a natural solution [duplicate]

let $n$ be a natural number and $r$ , $s$ be rational such that $n=s^2+r^2$ show that there are natural numbers a,b such that $n=a^2+b^2$
0
votes
1answer
56 views

Does $\pi \left(\dfrac{x+y}{2}\right)=\pi\left(\sqrt{xy}\right)$ hold for infinitely many values of $x$ and $y$?

The problem is (assume $\pi(x)$ to be the prime-counting function), Does there exist infinitely many solutions to the equality $\pi \left(\dfrac{x+y}{2}\right)=\pi\left(\sqrt{xy}\right)$ with ...
0
votes
1answer
70 views

Is the Riemann zeta function $\zeta(s)$ exactly $\pi(x)$?

Let $\pi(x)$ denote the number of primes less than or equal to a certain x value. The prime number theorem says that $x/\log x$ (or more accurately $x/(\log x-1)$) has been the most popular method ...
0
votes
1answer
41 views

Let p be an odd prime number. Then show the following:

Let $p$ be prime, $p \geq 3$. Then show that $K_p$ is the union of $\frac{1}{2}(p-1)C_p$. I am once again at a loss for a starting point. Maybe just a small hint so I can work through this myself ...
2
votes
1answer
49 views

p-th root does not become a p-th power when adjoined?

Suppose $k$ is a number field of characteristic zero, and $u$ is a unit of infinite order, which is not a $p$-th power in $k$. Show $\sqrt[p]{u}$ is not a $p$-th power in $k(\sqrt[p]{u})$. (You can ...
1
vote
0answers
22 views

Is there a function of, say, x and y that would take the first x factors in a factorial and return a xCy amounts of terms with y factors in each term?

What I'm basically looking for is described in the title. Here are some examples of what the function I'm looking for should do. Is there an existing function that does this? Even if not, are there ...
6
votes
1answer
83 views

Prove that $2AB$ is square [duplicate]

Let $$A= 1! \cdot 2! \cdot 3! \cdots 1002!$$ $$B= 1004!\cdot 1005! \cdots 2006!$$ Prove that $2AB$ is square. Help guys, I tried, I really did but I couldn't.
4
votes
1answer
54 views

For any $n$ positive integers ($n\geq 5$) exactly 3 or 4 of them are equal to each other modulo $2^m$ for some $m$

How can one prove that for any $n$ distinct positive integers, $n\geq 5$, there exists $m$ such that exactly 3 or 4 of them are equal to each other modulo $2^m$? I tried to prove it for small $n$. ...
5
votes
3answers
77 views

Can a square be in the form $2x + 1$, when $x$ is odd?

I was given this question, and I think I have solved it, but I'm not sure it is correct because this differs from how the answer is given. What is the most common way to solve this problem? Let's ...