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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4
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1answer
97 views

Integers with square-related properties

For the sake of convenience, let's define an integer to be a "supersquare" if: The number itself is a positive square number Each digit of the number is a positive square (1, 4, 9) The sum of digits ...
2
votes
0answers
86 views

Riemann-Weil formula question

given a sum over the imaginary part of the zeros $ \sum_{t} h(t) $ is this version of the Riemann-Weyl formula correct? $ \sum _{t} h(t) -2h(i/2)= -2 \sum_{n=1}^{\infty}\Lambda (n)g(\log ...
1
vote
3answers
62 views

Determine a linear modulo function's parameters

In a linear function without modulo like $y=ax+b$, if we know two points of $(x,y)$, the parameters $a$ and $b$ can be easily derived. But in the case of modulo linear function: $y=ax+b \pmod n$, ...
12
votes
3answers
414 views

Can $2^n$ and $2^{n+1}$ have the same digit sum?

For $n\in\mathbb{N}^*$ we note $s(n)$ the sum of the digits of $2^n$. For example $s(4)=7$ because $2^4=16$. Is it possible to find an integer $n$ such that $s(n)=s(n+1)$?
2
votes
1answer
123 views

Find all integers $n\in\mathbb{N}^*$ such that any integer $k$ satisfying $1\le k\le \sqrt{n}$ divides $n$

I've found following: 1, 2, 3, 4, 6, 8, 12, 24 and suspect that no integer larger than 24 satisfies the requirements. How do I prove that or can you find a counterexample?
11
votes
1answer
286 views

Is there any theoretical indication that this conjecture of Catalan could be true?

Belgian mathematician Catalan in $1876$ made next conjecture: If we consider the following sequence of Mersenne prime numbers: $2^2-1=3 , 2^3-1=7 , 2^7-1=127 , 2^{127}-1$ then $$2^{2^{127}-1}-1$$ is ...
2
votes
1answer
76 views

“Closed” form for $\prod\limits_{i=1}^{n} i^{[(i,n)=1]}$?

How could we derive a closed form for $\prod\limits_{i=1}^{n} i^{[(i,n)=1]}$? Here "$[s]$" is the Iverson bracket and "$(a,b)$" is the greatest common divisor.
1
vote
1answer
596 views

Perfect cubes and Fermat's Last Theorem

Question: If $n$ is a nonnegative integer, prove that $n + 2$ and $n^2 + n + 1$ cannot both be perfect cubes. Possible solution: Suppose $n+2$ and $n^2 + n + 1$ are perfect cubes, their ...
3
votes
0answers
60 views

Green’s formula in p-adic integration

Is there an analogue of Green's formula in p-adic integration (with respect to the Haar measure)?
2
votes
0answers
96 views

Is there a $\mathbb{C}_g$, where $g$ is composite?

Analogous to the $p$-adic ring $\mathbb{Z}_p$, you can (at least formally), define the $g$-adic ring $\mathbb{Z}_g$, where $g$ is composite. Of course when completing to a field, you get in trouble ...
15
votes
2answers
215 views

Find smallest number $n\gt 1$ for which the sum of the $q$th power of its digits is $n$

Today I came across this problem: For a given integer $q$, find the smallest natural number $n > 1$ such that sum of the $q$th powers of its digits is equal to $n$. For example, we can't find ...
3
votes
2answers
207 views

Split prime in $\mathbb{Z}[\sqrt{14}]$

I have this assertion: if $p$ is a prime such that $p\equiv 11 \pmod{56}$, then $p$ splits in $\mathbb{Z}[\sqrt{14}]$ (the discriminant of $\mathbb{Z}[\sqrt{14}]$ is $56$.) Why? Does $p\equiv ...
13
votes
2answers
418 views

Consequences of the Langlands program

I have been reading the book Fearless Symmetry by Ash and Gross.It talks about Langlands program, which it says is the conjecture that there is a correspondence between any Galois representation ...
3
votes
1answer
125 views

Easy system in five equations

An interesting little problem: I have found solutions $[a=3,b=2,c=1,d=5,e=4]$ but not able to find proof that these are all that exist. Find, with proof, all integers $a$, $b$, $c$, $d$ and $e$ such ...
6
votes
0answers
280 views

Certain permutations of the set of all Pythagorean triples

The fact that the set of all primitive Pythagorean triples naturally has the structure of a ternary rooted tree may have first been published in 1970: http://www.jstor.org/stable/3613860 I learned ...
3
votes
4answers
2k views

Prove that the set of integers are infinite

How can you prove that the set of integers are infinite? And can the proof be generalized to prove the set of natural numbers, rational numbers, and complex numbers are infinite?
5
votes
4answers
267 views

Quick proof to showing that $p\equiv 1\pmod{4}$ implies $p$ is reducible in $\mathbb{Z}[i]$?

I want to show that for $p$ a prime, if $p\equiv 1\pmod{4}$, then $p$ is not irreducible in $\mathbb{Z}[i]$. I know that $x^2\equiv -1\pmod{p}$ has a solution when $p\equiv 1\pmod{4}$, so $p\mid ...
1
vote
1answer
101 views

how many embeddings into $\overline{\mathbf{Q}}$ does a given number field have

Fix an algebraic closure $\overline{\mathbf{Q}}$ of the rational numbers. Let $\mathbf{Q}\subset K$ be a number field. I know that the degree $[K:\mathbf{Q} ]$ equals the number of embeddings of ...
4
votes
3answers
675 views

How to approach number guessing game(with a twist) algorithm?

I posted this on stackoverflow, but was advised to also post here. It's kind of a math/algo question so I think it's kind of stuck between both worlds of math and computer science. I believe this to ...
4
votes
2answers
426 views

Why does the natural ring homomorphism induce a surjective group homomorphism of units?

I'm trying an old problem here: http://www.math.dartmouth.edu/archive/m111s09/public_html/homework-posted/hw1.pdf Suppose $n\mid m$, and I have a natural ring homomorphism $\varphi\colon ...
12
votes
3answers
495 views

Twin primes of form $2^n+3$ and $2^n+5$

How to prove that $2^n+3$ and $2^n+5$ are both prime for only finitely many integers $n$? And how to prove that there are infinitely many primes of the form $2^n+3$ and $2^m+5$
5
votes
1answer
96 views

Primes of the form $p_{i_1}p_{i_2}\cdots p_{i_n}+2k$

Let $S_{n,k}$ be the set of all numbers that can be written as the product of $n$ odd primes plus $2k$. Is there integers $n>1$ and $k>1$ such that $S_{n,k}$ contains finite number of primes?
0
votes
2answers
322 views

Summing up all $N$ digits automorphic numbers

In mathematics an automorphic number (sometimes referred to as a circular number) is a number whose square "ends" in the same digits as number itself. Thus $5$ is automorphic since $25$ ends in ...
3
votes
1answer
297 views

Is this a good proof of Wilson's theorem? — ($(n-1)!+1 \equiv_n 0$ iff n is prime)

Theorem: $(n - 1)! + 1 \equiv_n 0$ if and only if $n$ is prime. To prove that if $n$ is not prime this is not true is trivial, so I'm just interested in proving that this is true for all p: ...
18
votes
2answers
915 views

Why do we use this definition of “algebraic integer”?

A number is an "algebraic integer" if it is the root to a monic polynomial with integer coefficients. Artin says (Algebra, p. 411): The concept of algebraic integer was one of the most important ...
11
votes
2answers
419 views

Asymptotics of LCM

Let $\operatorname{LCM}(x_1,x_2,\ldots,x_n)$ be the least common multiple of the integers $x_i$. How can one find the asymptotics of $\operatorname{LCM}(f(1),f(2),\dots,f(n))$ as $n$ approaches ...
3
votes
3answers
1k views

Riemann Hypothesis and prime number distribution

I do not grasp all concepts of the Riemann Hypothesis (better yet: as a layman I barely grasp anything...). However, I understand that there is a certain link between the Riemann Hypothesis and prime ...
6
votes
1answer
141 views

Trajectories on the $k$-dimensional torus

Let $r_1,\dots,r_k$ be irrational and linearly independent over $\mathbb Q$. My intuition clearly tells me that the set $$\{(nr_1,\dots,nr_k)+\mathbb Z^k:n\in\mathbb N\}$$ is dense in $\mathbb ...
28
votes
4answers
4k views

Would a proof to the Riemann Hypothesis affect security?

If a solution was found to the Riemann Hypothesis, would it have any effect on the security of things such as RSA protection? Would it make cracking large numbers easier?
1
vote
1answer
121 views

Do these zeros have real part equal to $0$?

I guess this is a known result but I could not find it on the Internet. Consider these equations formed from the reciprocals of the divisors of $n$ raised to a complex number $s=a+ib$ : ...
12
votes
2answers
741 views

What is the simplest ellipse that goes through exactly 13 lattice points?

The ellipse $-30 x + 3 x^2 - 10 y - 3 x y + 4 y^2$ goes through exactly 11 lattice points. Another such ellipse is $4 - 30 x + 2 x^2 - 5 y - x y + 3 y^2$. What is the simplest ellipse that goes ...
20
votes
2answers
829 views

Complex solutions for Fermat-Catalan conjecture

The Fermat-Catalan conjecture is that $a^m + b^n = c^k$ has only a finite number of solutions when $a, b, c$ are positive coprime integers, and $m,n,k$ are positive integers satisfying $\frac{1}{m} + ...
3
votes
2answers
322 views

Primes and proofs

1) Are there infinitely many primes of the form $a_n$? if $p_1 = 2 < p_2 = 3 <\cdots$ is the sequence of primes then are there infinitely many $n$ for which $p_1p_2\dots p_n + 1$ is prime? For ...
0
votes
1answer
92 views

Does this sum have an upper bound?

If we have an infinite sequence of positive numbers whose sum is $$ S = \sum_{i=1}^\infty a_n $$ and $$ \lim_{n \to \infty} a_n = 0 $$ Can we draw conclusion that $S$ has an constant upper ...
1
vote
2answers
362 views

Ratio of primes

How can one find the limit as M approaches infinity of the ratio of the number of primes p to the number of primes q all less then M. Where every p satisfy: p+42 is prime, and p+20 is prime. And ...
19
votes
2answers
326 views

All positive integer solutions to $\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_n}+\frac{1}{x_1 x_2 \cdots x_n}=1$

As the title states, how would I go about finding the positive integer solutions of $$\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_n}+\frac{1}{x_1 x_2 \cdots x_n}=1$$? Thank you for your help. ...
4
votes
3answers
117 views

Given $a$, $b$, and a prime $p$, how fast can we solve $(a \cdot c) - (b \cdot d) \equiv 1 \bmod p$?

If we're given two naturals, $a$ and $b$, and a prime $p$, how fast can we find two more naturals such that $(a \cdot c) - (b \cdot d) \equiv 1 \bmod p$? Additionally, you are allowed to precompute ...
6
votes
2answers
367 views

Puzzle: Can arithmetic be axiomatized with a single two-term relation?

Following my question about defining multiplication in terms of divisibility, can all of arithmetic be axiomatized with a single two-term relation? Asaf Karagila comments on my question that the ...
3
votes
1answer
88 views

Totally ramified cyclic extensions of degree $p^a$ of $\mathbb{Q}_p$

It's quite easy to show that the totally ramified extension $\mathbb{Q}_p(\zeta_{p^{a+1}})/\mathbb{Q}_p$ contains a unique subextension $E$ s.t. $E/\mathbb{Q}_p$ is a cyclic extension of degree $p^a$ ...
10
votes
2answers
215 views

Probability p+k is a prime

If p is a prime number, and k is an even integer, what is the probability p+k is a prime number? According to my simulations p+108 is prime twice as often as p+344
6
votes
2answers
263 views

representing all odd naturals as the sum of four squares, two of them equal

can anyone prove that every natural odd number can be represented as the form $a^2+b^2+2c^2$ where $a,b$ and $c$ are nonnegative integers? I've thinked on this problem for a long time, but I couldn't ...
7
votes
4answers
354 views

Is there a procedure to determine whether a given number is a root of unity?

Let $z$ be an algebraic number of modulus one. Is there a finite procedure that tells us whether $z$ is a root of unity? EDIT: As TonyK and David asked, what I had in my mind is $z$ such that I have ...
16
votes
2answers
494 views

Why is $f(x) = x\phi(x)$ one-to-one?

I noticed that $f(x) = x\phi(x)$ seems to be one-to-one, where $\phi(x)$ is Euler's Phi function. In particular, I'm writing some numerical python code and the line I have looks something like ...
5
votes
0answers
195 views

Sums of Dirichlet-Characters over prime numbers (part 2)

This is kind of related to my previous question that was poorly stated because of misreading my own notes that I have taken on the papers I am currently reading, so no surprise that it eventually ...
-5
votes
2answers
2k views

Is the Birch and Swinnerton-Dyer conjecture solved?

I read today that in 2010 Manjul Bhargava with Arul Shankar proved the conjecture basing upon the work of Kolyvagin. Is it right? Does it satisfy for all elliptic curves, or is it limited to some ...
5
votes
1answer
96 views

Shifting the line of integration over the critical stripe of an L-function

Let $F$ be an L-function in the sense of the Selberg class --> http://en.wikipedia.org/wiki/Selberg_class. We are observing the integral $$\frac{1}{2\pi i}\int_{(c)}F(s+w)\Gamma(w)z^w dw$$ for ...
7
votes
1answer
193 views

Asymptotics of sums of Dirichlet-Characters over prime numbers

Again in relation with some stuff I am currently reading, the authors make use of the following "standard argument in prime number theory": Let $\chi$ be a non-principal Dirichlet-character. Then ...
1
vote
1answer
187 views

Tamagawa number conjecture

I heard somewhere that the above formulation of conjecture is for predicting the exact leading term of a L-function at an integer. But i didnt find any reference about how it is stated, anyone please ...
4
votes
3answers
5k views

Special numbers

Our teacher talked about some special numbers. These numbers total of 2 different numbers' cube. For example : $x^3+y^3 = z^3+t^3 = \text{A-special-number}$ What is the name of this special numbers ...
5
votes
2answers
333 views

How can I reduce a number?

I'm trying to work on a program and I think I've hit a math problem (if it's not, please let me know, sorry). Basically what I'm doing is taking a number and using a universe of numbers and I'm ...