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.

learn more… | top users | synonyms (1)

5
votes
1answer
148 views

What is the analytic continuation of a multifactorial?

The $\Gamma$ function is the analytic continuation of the factorial function. Is there a similar analog for multifactorials? I am particularly interested in the double factorial. All Google has ...
3
votes
3answers
234 views

Comparing $\large 3^{3^{3^3}}$, googol, googolplex

How to show that $\large 3^{3^{3^3}}$ is larger than a googol ($\large 10^{100}$) but smaller than googoplex ($\large 10^{10^{100}}$). Thanks much in advance!!!
-1
votes
1answer
91 views

primes of type norm($a+b\sqrt{-c}$) = primes of type $1$ $mod$ $c 2^{n-1}$?

Let $a,b,c,n$ be strictly positive integers where $c$ is a prime and such that the normed ring $a+b\sqrt{-c}$ is a UFD. I noticed the primes of norm($a+b\sqrt{-1}$) are $1 \bmod 4$. Also the ...
0
votes
1answer
78 views

primes of the form $a^2+b^2=x^2-xy+y^2$?

Let $a,b,x,y$ be strict positive integers. Im intrested in primes $p$ such that $p=a^2+b^2=x^2-xy+y^2$. What is the analogue PNT for these type of primes ? I think these primes are all the primes $p ...
1
vote
1answer
317 views

Converse of Collatz Conjecture

How to write a pseudocode program that halts only if the Collatz Conjecture is false. Thanks much in advance!!!
0
votes
2answers
222 views

Congruence class's proof of with 0 divisors

Question Let n be a positive integer and consider $[a]$ in $Zn$. Prove that $[a]$ is a zero divisor in $Zn$ if and only if it does not have an inverse in $Zn$. Bad proof. Theorem An element $[a]$ ...
5
votes
5answers
357 views

Show that there are no nonzero solutions to the equation $x^2=3y^2+3z^2$

I am asked to show that there are no non zero integer solutions to the following equation $x^2=3y^2+3z^2$ I think that maybe infinite descents is the key. So I started taking the right hand side ...
1
vote
1answer
127 views

Isogeny and minimal models of elliptic curves

Suppose I have two isogenous elliptic curves over $\mathbb{Q}$, $E$ and $E'$. Will the minimal models of $E$ and $E'$ still be isogenous?
1
vote
1answer
62 views

Help with proof : $Cl_F$ with only one equivalance class implies $D_F$ is a PID

I am self-studying a set of legitimately downloaded notes on algebraic number theory. They are somewhat akin to "Ireland and Rosen," Ch. 12. I would appreciate help in understanding a proof (in the ...
1
vote
1answer
103 views

Application of the fundamental theorem of algebra

I'm just going through my lecture notes and under one subheading "application", I have written: If $m = \prod_{i = 1}^k p_i^{r_i}$ and $n = \prod_{i = 1}^k p_i^{s_i}$, for primes $p_i$ & $r_i, ...
3
votes
4answers
7k views

Is this set closed under addition or multiplication or both and why?

$\{-1,0,1\}$ Please give an explanation and also tell me what does closed under addition and multiplication mean. Different definitions are given everywhere.
4
votes
1answer
162 views

Discriminant of isogenous elliptic curves

Let $E$ be an elliptic curve with a rational $p$-torsion point $P$. Then $E$ is isogenous to the elliptic curve $E' := E/\langle P \rangle$ via the mod $P$ map. I know that the conductor of $E$ and ...
1
vote
2answers
152 views

Finite fields are isomorphic

This is from A Course in Arithmetic by JP Serre Theorem 1 ii) Let $p$ be a prime number and let $q = p^f(f \geq 1)$ be a power of $p$. Let be an algebraically closed field ...
5
votes
3answers
164 views

prime numbers and some conjectures

Consider triples $(p,q,r)$ of prime numbers $p$, $q$ and $r$ such that $(p+1)(q+1)=(r+1)$. Here are some examples : $(2,3,11), (3,7,31)$. How to prove there are infinitely many such triples?! I ...
2
votes
2answers
190 views

How can I find the integer solutions to $x^2+x-2y^2=0$?

I enter this equation in Wolfram Alpha : $x^2+x-2y^2=0$ and it gave me something like this : and I am wondering how this solution is found and how to know if a given equation would guarantee to ...
10
votes
5answers
830 views

Is $\mathbb Z _p^*=\{ 1, 2, 3, … , p-1 \}$ a cyclic group?

In undergraduate course, the two groups which are most frequently used may be $$\{ 0, 1, 2, ... , p-1\}$$ and $$\{ 1, 2, ... , p-1\}$$ where $p$ is a prime. The first one is a group under addition ...
1
vote
2answers
102 views

Sequence of integers

This one should be quite easy, I tried with mathematical induction but the things started to complicate so I would like to see how someone of you there will prove this: Prove that ...
4
votes
1answer
122 views

Prime divisors of $n^{3} - 27$ and squarefreeness

Let $n \equiv 4 \bmod{6}$. Does there exist infinitely many $n$ such that $3 \nmid \operatorname{ord}_{p}(n^{3} - 27)$ for each prime $p \mid n^{3} - 27$? In particular, the following (possibly ...
0
votes
2answers
273 views

Fermat Numbers as a product

We are discussing Fermat numbers in class, and one of the claims brought up is as follows: "For any integer $n \ge 1$, the $n$th Fermat number is $F(n)$ = $2 + \prod_{i=0}^{n-1}F(i)$." I have not ...
1
vote
2answers
47 views

Explaining how $n = 2^r$ for $n$ prime

I found this claim in my textbook while reading the section on prime numbers today: "If $n$ is a positive integer such that $2^n + 1$ is prime, then $n = 2^r$ for some integer $r \ge 0$." Where did ...
2
votes
0answers
119 views

Imposing condition of specification of product of $n$ of imaginary numbers on coefficients of imaginary numbers

I asked the same question but with some fatal mistake that makes the question unanswerable - so I decided to delete it and start new. Connecting from The set of numbers that when multiplied do not ...
5
votes
2answers
201 views

Classifying mathematical “coincidences”

Doing homework a few years ago, I noticed that the sum of the squares of $88$ and $33$ is $8833$. What would this kind of mathematical "curiosity" be called? Does this or any other similar ...
0
votes
3answers
46 views

Showing if I choose a set of $10$ unique numbers from $0$ to $14$

How would I show if I choose a set $10$ unique numbers from $0$ to $14$, there exists two numbers in the set such that their sum is greater than the largest number in the set?
1
vote
1answer
104 views

Primes of the form $n^{3} + 2$

Is it known that there are infinitely many primes which can be represented by $n^{3} + 2$ (or similarly any cubic polynomial)?
2
votes
1answer
175 views

Proving the condition for two elliptic curves given in Weierstrass form to be isomorphic

I'm taking a course on elliptic curves and trying to understand the proof of Proposition 3.2. Let $E$, $E'$ be elliptic curves over $K$ in Weierstrass form: ...
4
votes
1answer
110 views

An equation concerning eisenstein integers

An interesting exercise on gaussian integers is to prove that those of the form $n-i$, with $n$ a positive integer, are multiplicatively independant. To solve this, one has to consider the equation ...
0
votes
1answer
170 views

Formal identity for sum of polynomials over a finite field.

Suppose $F$ is a finite field of order $q$ a prime power. If $f\in F[x]$ of degree $t$, set $|f|=q^t$. Let $\sigma(f)=\sum_{g\mid f}|g|$ where the sum is over the monic divisors of $f$. Why does ...
1
vote
3answers
105 views

bound for the product of numbers

Let $n \in N$. Fix $m \in [-n,n]$. I am curious, how to bound from above the following expression $$ (n-m)^{\frac{n-m}{2}+1}(n+m)^{\frac{n+m+1}{2}}\leq \quad ? $$ Thank you.
50
votes
5answers
4k 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 ...
1
vote
1answer
95 views

Differences among different sieves encountered in sieve theory

Are the sieve techniques used in understanding the twin prime conjecture or other number theoretical conjectures different from sieve theory used in primality testing or integer factorization? What ...
2
votes
0answers
109 views

Diophantus again; not to say Pell.

Is there a way to solve the second degree Diophantine equation in two variables $ax^{2} -ny^{2} = b$ $(1)$ where a and b are known and n is a parameter; all solutions x= f(n) and y = f(n) ? For ...
1
vote
0answers
73 views

Any work on properties of $N + \bar \phi (N)$?

I am looking for pointers to any existing materials about the properties of this quantity. For Euler's cototient, if a number $N$ is written as $2^a \cdot b$ with b odd then the cototient is $$\bar ...
3
votes
2answers
89 views

If $2^k -1$ is a perfect square, do we have more than one solution?

I am trying to solve the equation $2^k-1 = x^2$, I have got one solution $k = 1$. How to proceed further i.e. either show that the equation has no more solutions or has more. Thanks
7
votes
2answers
621 views

Resource for Vieta root jumping

I can't seem to find a good resource on Vieta's root jumping, which would explain various scenarios that suggest using the technique. Does anyone have a suggestion for a reference?
3
votes
1answer
82 views

Generators for $\mathbb F_p^*$

Let $p$ be prime, then it is a well-known fact that $\mathbb F_p^*= \mathbb F_p -\{0\}$ is a cyclic group under multiplication. Are there any methods to determine the generators of this cyclic or any ...
1
vote
4answers
242 views

Find infinitely many pairs of integers $a$ and $b$ with $1 < a < b$, so that $ab$ exactly divides $a^2 +b^2 −1$.

So I came up with $b= a+1$ $\Rightarrow$ $ab=a(a+1) = a^2 + a$ So that: $a^2+b^2 -1$ = $a^2 + (a+1)^2 -1$ = $2a^2 + 2a$ = $2(a^2 + a)$ $\Rightarrow$ $(a,b) = (a,a+1)$ are solutions. My motivation ...
5
votes
1answer
127 views

Gauss Sum of a Field with Four Elements

I need to calculate a couple of Gauss sums to solve a problem I'm working on, but I keep getting the wrong answer because the absolute value of what I calculate is impossible for such a sum. Can ...
1
vote
0answers
172 views

Pentagonal-Triangular numbers

Pentagonal Triangular Number is a number which is simultaneously a pentagonal number $P_n$ and triangular number $T_m$ . Such numbers exist when $$ \frac{1}{2}n(3n-1) = \frac{1}{2}m(m+1) $$ This ...
2
votes
0answers
101 views

I have the following Diophantine equation $x=y^2-z^2$ [duplicate]

Possible Duplicate: $a^2-b^2 = x$ where $a,b,x$ are natural numbers Show that for every positive odd integer $x$ there exist integers $y$ and $z$ such that $(x, y, z)$ is a solution ...
4
votes
0answers
457 views

Defining the Riemann-Roch space of a divisor

I'm doing a course on elliptic curves. It starts with a bit of a crash course in algebraic geometry, giving statements alone. We were given the following definition The Riemann-Roch space of $D$ ...
3
votes
1answer
198 views

Hadamard's product of Riemann's $\xi$ function

I can prove that the order of $\xi(s)$ is at most 1. Therefore, from Hadamard's factorization theorem, it follows that $$\xi(s)=Ae^{Bs}\prod_\rho{ \left(1-\frac{s}{\rho}\right)e^{s/\rho}}$$ where A,B ...
2
votes
3answers
468 views

number of integers having at least one even digit

if we have a range specified (say $[A,B]$) then how can we find efficiently the number of integers in this range which have at least one even digit $(0,2,4,6,8)$? One way would be to iterate through ...
0
votes
1answer
265 views

Why the Wu-sprung model is not accepted as a solution to Riemann HYpothesis ??

in the Wu-sprung model, given a Hamiltonian in one dimension $$ -y''(x)+f(x)y(x)=E_{n}y(x) \qquad y(0)=0=y(\infty) $$ we can define the function $ f(x) $ implicitly as $$ f^{-1}(x)= 2\sqrt{\pi} ...
0
votes
2answers
203 views

Find square root of non-rational fraction

If we have to compute this without using calculator, is there a quick way to find the answer approximately of the following problem: which one is smaller ? $$ A = ...
15
votes
4answers
427 views

sum of cubes of two rationals

How to find two rational numbers $x,y$ such that $$x^3+y^3=6$$ I know that $x=17/21,y=37/21$ is a solution but I am interested in a method how is achieved and does exists other solutions
0
votes
1answer
47 views

The set of numbers that when multiplied do not get decomposed into $sx+ty$ while the numbers themselves are of form $ax+byi$

Suppose that there are $n$ numbers that are in the following format: $a+bi$. Each number has different combination of $a$ and $b$. $a,b$ must be non-zero integers. Suppose that we impose the ...
13
votes
2answers
345 views

When is $(p - 1)! + 1$ a power of $p$?

A friend asked me this question: If $p$ is a prime, prove that $(p - 1)! + 1$ is a power of $p$ if and only if $p = 2, 3$ or $5$. Clearly one direction is obvious, namely that $p=2,3,5$ implies ...
2
votes
3answers
282 views

Why the zeta function?

Why is the zeta function, $\zeta(s)$ used to obtain information about the primes, namely giving explict formula for different prime counting functions, when there are many other functions that encode ...
1
vote
1answer
116 views

Is the sum always bigger than $n^2$?

Let $s(n)$ an arithmetical function defined as $$s(n)=(p_1+1)^{e_1} (p_2+1)^{e_2} \cdots (p_m+1)^{e_m}$$ where prime factorization of $n$ is $n=p_1^ {e_1} p_2 ^{e_2} \cdots p_m^{e_m}$. (For example, ...
3
votes
1answer
160 views

A problem in prime number theory

I was wondering if anybody here might provide me with a hint for this rather innocuous-looking problem: If $X:= \{pq: p, q \mbox{ are prime numbers and } p\neq q\}.$ In addition, let us suppose that ...