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
11 views

Concept of combinations and number theory.

The number of six digit numbers of the form ababab(in base ten)each of which is a product of exactly six distinct primes?
1
vote
1answer
37 views

Determining parity of the multiplicative inverse?

Let $\mathbb{F}_p$ be a finite field of characteristic $p > 2$, for a fixed $p$. I will consider only prime fields, not $GF(p^n)$. Represent the $p$ elements of the field as integers $\{0,1,\ldots ...
3
votes
0answers
38 views

Approximate a large number with perfect powers

I'm dealing with number theory now and I have an interesting question. Every number can be approximated with two perfect powers, where perfect power is a number in form $$a^b$$$$a,b \geq 2, a,b \in ...
0
votes
0answers
325 views

Can this we define the zeta function like this?

Background We define a smooth continuous function function where: $$ p(i) = p_i $$ where $p_i$ is the i'th prime. We also define the following series: $$ \alpha(s) = (\ln(2))^s + (\ln(3))^s + ...
5
votes
1answer
34 views

$F[[T]] \times F[[1/T]]$, fundamental domain.

Let $p$ be a prime number. Here is a link which shows how to see that $$(\mathbb{F}_p((T)) \times \mathbb{F}_p((1/T)))/\mathbb{F}_p[T, 1/T]$$is compact using an adelic result. (Here $\mathbb{F}_p[T, ...
1
vote
1answer
56 views

What the difference between the smallest two numbers from these numbers?

There are infinitely many integers $n$ bigger than $1$, such that if we divide $n$ by any integer $k$ where $2\leq k\leq 11$, the remainder is equal to $1$. What the difference between the smallest ...
-1
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0answers
28 views

How to compute $Z_n \times Z^*_m$

How to compute $Z_n \times Z^*_m$.(In journal Multiplicative Properties of Set Residues) say by chinese remainder theorem, may be thought of as $\left\{a \in Z_{mn}:(a,m)=1\right\}$. where $Z^*m$ unit ...
-1
votes
0answers
21 views

Diophantine Equation with gcd. [duplicate]

Find all positive integers $a,b$ such that $\gcd(3^a+1,3^b+1)$ is a multiple of $ab$. I've given this problem many attempts but I can't seem to make any progress, there doesn't appear to be any way ...
-3
votes
1answer
42 views

solve $(x-3)^2 + (x+1)^2 + (4x-5)^2=0$

solve $(x-3)^2 + (x+1)^2 + (4x-5)^2=0$ this is what I have tried $$(x-3)^2=(x+1)^2=(4x-5)^2=0$$ $$x=3, x=-1, x=\frac{5}{4}$$
0
votes
3answers
90 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 ...
4
votes
1answer
59 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, ...
1
vote
0answers
40 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
votes
0answers
25 views

What interesting results can be drawn from this expression?

What interesting results can we draw from this expression? $$\zeta(s)=\left(1+\sum_{r>0}\sum_{\{r\}}\prod_{j=1}^t \frac {(-1)^{i_j}}{{i_j}!}\left(\frac {P({r_j}\cdot ...
6
votes
0answers
41 views

$y^2 = x^3 - 26$, exist ideal satisfying conditions?

For the solution $(x, y) = (3, 1)$ of $y^2 = x^3 - 26$, does there necessarily exist an ideal $I$ of the integer ring $\mathbb{Z}[\sqrt{-26}]$ of $\mathbb{Q}(\sqrt{-26})$ such that $(y + \sqrt{-26}) = ...
16
votes
0answers
221 views
+100

Meromorphic functions on $U^2 = T^3 + 1$, genus.

Let $k$ be a field of characteristic $\neq 2$, and consider the quadratic extension $F$ of $k(T)$ generated by $\sqrt{T^3 + 1}$. What is/how do I find the genus of $F$? The progress I have so far: ...
0
votes
2answers
95 views

problems on congruent number generating and others…

From where we got this congruent generating function $n = pq(p+q)(p-q)$? how to prove 1- is non-congruent number? Thanks in advance
3
votes
2answers
109 views

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

Show that $$a^7+b^7=7^c$$ has no positive integer solutions $(a,b,c)$. I've posted a general and way too long approach as an answer. How may one prove the claim more briefly and specifically?
1
vote
1answer
13 views

Imprimitive Dirichlet Characters

I've started to read the fifth chapter of "Multiplicative Number Theory" by Harold Davenport and I got stuck at some point. Let me elaborate the part that i didn't quite understand. Let $\chi$ be ...
0
votes
0answers
29 views

For what $n$ can this sum be an integer? [duplicate]

Consider the well known $\sum_{k=1}^{n} \frac{1}{k}$ sum. My question is simple: How can we choose $n$ in order to make the sum integer? My approach: The first obvious solution is $n=1$. I tried ...
9
votes
1answer
285 views

On Grunwald-Wang theorem

Consider (roughtly speaking) the following statements (the Grunwald-Wang theorem) Theorem 1 (see here for details Wiki): Let $K$ be a number field and $x \in K$. Then under some conditions : $x$ is ...
-1
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0answers
28 views

How does Graham knows his number is really the upper bound to the dimension problem?

I know initially he stated that the answer is somewhere between 6 and Graham's number. How does he know that for Graham's number dimensions it is really impossible to color the lines that way? I know ...
3
votes
1answer
137 views

Solve $x^n+z^n=(x+1)^n$ for $n\ge 3$ without FLT

Is there a way to prove that for $x,z,n \in \mathbb{Z}$, $x > 0$, $z > 0$, $n > 2$, the equation $$ x ^ n + z ^ n = (x + 1) ^ n $$ has no solution, without using Fermat's Last Theorem? ...
6
votes
1answer
3k views

The number of partitions of $n$ into distinct parts equals the number of partitions of $n$ into odd parts

This seems to be a common result. I've been trying to follow the bijective proof of it, which can be found easily online, but the explanations go over my head. It would be wonderful if you could give ...
2
votes
3answers
79 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 ...
2
votes
1answer
44 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 ...
6
votes
1answer
43 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 ...
2
votes
1answer
42 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 ...
1
vote
1answer
61 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 ...
5
votes
2answers
70 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 ...
7
votes
1answer
86 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 ...
13
votes
2answers
144 views

Numbers divide its prime factors' concatenation

Check these out! $$28749=3\cdot 7\cdot 37\cdot 37\; \text{ and } \;28749\mid 373737$$(amazing!). Even much more interesting than that is this number (because the digits in its prime factors looked ...
0
votes
0answers
15 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 ...
51
votes
6answers
5k views

Is the notorious $n^2 + n + 41$ prime generator the last of its type?

The polynomial $n^2+n+41$ famously takes prime values for all $0\le n\lt 40$. I have read that this is closely related to the fact that 163 is a Heegner number, although I don't understand the ...
4
votes
4answers
75 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 $p^4-1$ ...
1
vote
1answer
446 views

Sum of three primes equal to a prime [closed]

Does anyone know how to always get a prime from the sum of three primes? For example: $5+7+11=23$, $17+29+43=89$, etc.
0
votes
3answers
158 views

A question on consecutive prime numbers

Prime numbers: $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ .... Difference between to consecutive primes: $1$ $2$ $2$ $4$ $2$ $4$ $2$ $4$ $6$ .... We know that there are infinite prime numbers. ...
2
votes
3answers
3k views

First 10-digit prime in consecutive digits of e

Problem. What is the first $10$-digit prime in consecutive digits of $e$. For those of you who don't know, in 2004 the answer produced a URL to a Google employment page (sort of). I just found about ...
1
vote
1answer
125 views

Prime made from the digits of $\sqrt{22}$

Which is the smallest prime derived from the digits of $\sqrt{22}$, where the 4 before the comma is not considered ? To be more precise : $x:=\sqrt{22}-4$ , so $x = 0,690415...$ for every natural ...
8
votes
1answer
120 views

Number made from ending digits of primes

Consider the number $0.23571379391713739171393971379371799173739113791379391173917133713717793$ ... The number is formed by the ending digits of the prime numbers. Is it known whether this number ...
9
votes
2answers
485 views

Primes dividing the values of integer polynomials

Problem: Let $n$ be an integer and $p$ a prime dividing $5(n^2-n+\frac{3}{2})^2-\frac{1}{4}$. Prove that $p \equiv 1 \pmod{10}$. The polynomial can be re-written as ...
5
votes
1answer
60 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
28 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 ...
3
votes
3answers
328 views

Proof of a special case of Fermat's Last Theorem.

Here, I will try to prove a special case of Fermat's Last Theorem, namely when $a=b$ in this definition: Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called ...
25
votes
5answers
15k views

Is there a known mathematical equation to find the nth prime?

I've solved for it making a computer program, but was wondering there was a mathematical equation that you could use to solve for the nth prime?
1
vote
1answer
42 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; ...
3
votes
1answer
136 views

How to prove $p(z-y)(zy)(z^2-yz+y^2) \mid x^p-(z-y)^p \Rightarrow x=z-y$?

Assume $p>2$ prime and $1<x<y<z$ coprime. How to prove the following: $$p(z-y)(zy)(z^2-yz+y^2) \mid x^p-(z-y)^p \Rightarrow x=z-y$$ I remember it as an extra exercise which I couldn't ...
2
votes
3answers
76 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 ...
0
votes
0answers
16 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 ...
2
votes
1answer
78 views

For which integer $n$, $\sin\left(\frac{\pi}{n}\right)$ can be a rational?

When I studying the trigonometric functions, I sow that most of the values of $\sin\left(\dfrac{\pi}{n}\right)$ and $\cos\left(\dfrac{\pi}{n}\right)$ where $n\in\mathbb{N}$ are irrational. How can we ...