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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4
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1answer
44 views

if $g|ab$ , $g|cd$, $g|ac+bd$, show $g|ac$ and $g|bd$.

Struggling to solve this problem. Professor suggest we look at $p^n$ as one of the prime factorizations of $g$ (note $p^{n+1}$ doesn't divide $g$) and likewise the number of $p$'s in $a, b, c,$ and ...
7
votes
0answers
84 views

The smallest number that if multiplied by 2 forms a permutation of itself

I am looking for the smallest number larger than $0$ which when multiplied by $2$, forms a permutation of itself. I quickly remembered that the number $142,857$ does that, as well as with all numbers ...
0
votes
0answers
48 views

A Problem from Marcus' Number Fields

I have been stuck on the 17th problem of the 3rd chapter from Marcus' Number Fields. Let $K=\mathbb{Q}[\sqrt-23]$ , $L=\mathbb{Q}[\omega]$ where $\omega = e^{2.\pi.i/23} $ . Let $P$ be one of the ...
1
vote
1answer
82 views

Prime factorization of $\frac{3^{41} -1}{2}$ [duplicate]

How can I determine all prime factor of $\frac{3^{41} -1}{2}$ without doing prime factorizations. Any help will be appreciated.
-1
votes
0answers
43 views

If $p$ is congruent to 3 mod 4 then $2^p -1$ is not a prime

Let $p=3 \pmod 4$ be a prime number such that $q = 2p+1$ is also a prime number. Then I want to show that $q$ divides $2^p -1$. Thank you so much. Any help will be appreciated.
1
vote
1answer
59 views

There are infinitely many primes congruent to 9 mod 10

I want to show that there are infinitely many primes $p$ such that $p = 9 \pmod {10}$. First, I can see that 19 is one of them. Assume there are finitely many, i.e., 19, $p_1, p_2 , \cdots , p_k$. ...
4
votes
2answers
102 views

Without using prime factorization, find a prime factor of $\frac{(3^{41} -1)}{2}$

Not sure how to go about this. Law of quadratic reciprocity and Euler's Criterion is recently learned material but I'm not sure how this applies.
0
votes
1answer
23 views

Use the prime factorization of $\phi(321)$ to determine possible orders of units mod 321

Use the prime factorization of $\phi(321)$ to determine possible orders of units mod 321. (Your list should have fewer than ten numbers in it). My attempt: $\phi(321)$ = $\phi(3)\phi(107)$ = $2(3)$ ...
5
votes
0answers
35 views

Proving that $-1$ has no square root in $\mathbb Z_3$ (3-adic integers)

I want to prove that $-1$ has no square root in $\mathbb Z_3$ (3-adic integers). By the definition of inverse limit, we know that there is a ring homomorphism $\phi$ from $\mathbb Z_3$ to $\mathbb ...
1
vote
2answers
21 views

Integer sequences that satisfy the finite intersection property

I am curious what integer sequences are there that satisfy the "finite intersection property". To be precise, what are the family of integer sequences that any intersection of two such sequences in ...
2
votes
0answers
33 views

Is there anyway to find how many prime factors has a composite number without knowing them?

Let's call f(n) the function that gives us the number of different prime factors of a composite number n For example: f(24)=2 Let's call g(n) the function that gives us the number of prime factors of ...
4
votes
2answers
33 views

Primes of the form $(2p)^{2}+1$, $p$ prime, have $h^{2}+1$ as a prime divisor?

I'm an undergraduate student and I usually ask questions here about things I'm struggling with in my academical mathematical studies, but this particular question is actually more like a curiosity. ...
1
vote
1answer
20 views

divisibility theorem proof?

I have found in a book the proof for the divisibility problem that says: If $a$ and $b$ are integers and $b$ is not equal to zero, then there is a unique pair of integers $q$ and $r$ such that ...
0
votes
0answers
19 views

Interrelated sets or numbers

Consider the ordered collection of digits base $10$ of length $m, A=a_1a_2a_3...a_m$. Let us look at some forms of inter-relation in these numbers. Here is an example of interrelation. Let vicinity of ...
1
vote
0answers
24 views

The tower of ramification indices

Let $K\subset L\subset M$ be an extension of number fields. Let $R\subset S\subset T$, be their algebraic integers rings, respectively. Suppose that $P\subset R$, $Q\subset S$, $U\subset T$ be a prime ...
0
votes
0answers
22 views

Understanding Ziv's proof of zero sum problem .

I was going through the proof of zero sum problem in one dimension as provided by Abraham Ziv. The problem statement is to prove that given a set of $2n+1$ integers, we can find at least $n$ integers ...
12
votes
2answers
1k views

Is there an explicit irrational number which is not known to be either algebraic or transcendental?

There are many numbers which are not able to be classified as being rational, algebraic irrational, or transcendental. Is there an explicit number which is known to be irrational but not known to be ...
1
vote
1answer
45 views

Prove that there does not exist integer solutions for the diophantine equation $x^5 - y^2 = 4$.

Prove that there does not exist an integer solution for the diophantine equation $x^5 - y^2 = 4$. It's obvious that $x$ and $y$ are of the same parity. We can also claim that if $x$ is odd, then ...
0
votes
0answers
34 views

Does Inverse of Phi(N) Mod (N) always exist?

Let N=pq where p and q are odd primes and let $\Phi(N)$ be Euler Phi function i.e. $\Phi(7) = 6$ since 6 numbers co-prime to 7 Does the inverse of $\Phi(N) Mod(N^2)$ always exist? It exists if and ...
0
votes
0answers
59 views

Show that $1+X^4$ is reducible over $\mathbb Z_p$ for every prime $p$. [duplicate]

Show that $1+X^4$ is reducible over $\mathbb Z_p$ for every prime $p$. MY ATTEMPT==>I have used Fermat's theorem for this as $X^{p-1}≡1\bmod p$, then this can also be written in the form ...
-4
votes
2answers
142 views

Why is minimum solution example to $x^n + y^n = z^n$ comes in the form of three successive integers? [closed]

Can we prove or disprove this conjecture by elementary mathematics: If this is a true statement: $$x^n + y^n = z^n $$where $x, y, z, n$ are positive integers, then there must be a minimum integer ...
1
vote
1answer
54 views

Proof involving Fundamental Theorem of Finitely Generated Abelian Groups

Suppose $G \simeq \mathbb{Z}_{m_1} \times \cdots \times \mathbb{Z}_{m_t}$ where each $m_i$ is a positive integer (not necessarily prime). Let $p$ be a prime, and let $s$ be the number of $m_i$'s ...
2
votes
0answers
29 views

Stronger form of Hensel's lemma?

Let $f \in \mathbb{Z}_p[x]$ and suppose $|f(a)|_p < |f'(a)|_p^2$ for some $a \in \mathbb{Z}_p$. Let $a_1 = a$, and for $n \ge 1$ let$$a_{n+1} = a_n - f(a_n)/f'(a_n).$$How do I see that this defines ...
0
votes
1answer
75 views

Congruence - Number Theory

Prove that $2005^{2005}$ is not the sum of two perfect cubes. I have looked at some mods but none have given me anything useful as of yet. I looked at the usual mods such as $4, 5, 7, 11, 13$ but ...
-4
votes
1answer
17 views

What's the basic steps to show a set is denumerable? [closed]

For example, $\mathbb{N}$ is denumerable.
1
vote
0answers
16 views

Lower bound on $\pi(x/2)$

I have seen the bound, $\pi(x/2)^2\gg\frac{x^2}{\log^2x}$ (In particular here http://staff.polito.it/danilo.bazzanella/PhD_files/Not%20always%20buried%20deep%20(Pollack).pdf page 212) Can someone ...
0
votes
0answers
19 views

Number theory questio [duplicate]

I have studied for about 250 questions and there are many questions i can't take.. i considered but i don't know how to start... how can i solve this? In second question, there even hint.. Well ...
1
vote
1answer
35 views

is this question Using chinese remainder theorem?

I think that it will use Chinese remainder theorem but I don't know how to put... well by CRT, there exist $x=k\mod (m_1)(m_2)(m_3)$ which is $x=a_1^3 \mod m_1$ and $x=a_2^3 \mod m_2$ and ...
0
votes
0answers
23 views

Properties of multiplying in quadratic residue with composite modulo

I know the basic properties, theories and their proofs in quadratic modulo when the modulo is prim. (Euler's theory, Gauss lemma, using Legendre and Jacobi symbols..) When the modulo is an odd prim ...
-1
votes
1answer
28 views

number theory.. fibonacci

i don't know where to start... above question is... i think that it will using crt. like x=> x1^3=a (modp) lifting x=a (modp^2) x=> x1^3=a (modq) lifting x= (modq^2) and by crt, ...
-1
votes
0answers
21 views

Unique factorization property 3i [duplicate]

I know about unique factorization And i think that it will be used in these kinds of question that Norm But i want to know the way of proving the K has unique factorizatiob property If it doesn't ...
0
votes
2answers
25 views

How to test qualities about a number's prime factorization in maple?

I'm trying to write a code to test the following properties of a composite number $n$. To be more specific, for a given $n$ from (say) 200-300, I want to check if $n$ satisfies any of the following: ...
1
vote
1answer
26 views

Finding polynomial in modular? // $(n!)+1$ prime

Questions: Find a polynomial $f(x)$ satisfying $$f(x) \equiv x^2 + 3x + 4 \pmod {13}$$ $$f(x) \equiv 7x + 1 \pmod {17}$$ Is $(14!)^2+1$ prime or not? Give your full reasoning. Well I ...
0
votes
1answer
36 views

Show that if a quadratic form is primitive then so are equivalent forms

A Quadratic form is primitive if the greatest common divisor of the coefficients of it's terms is 1. I saw in number theory book that "it is easily seen that any form equivalent to a primitive form ...
1
vote
0answers
60 views

The sum of the greatest common divisors

What are the values for the positive numbers $a,b$ and $c$ can take the expression $$(a^2,b^2)+(a,bc)+(b,ca)+(c,ab)?$$ (Here $(u,v)=\gcd(u,v)$ - the greatest common divisor for $u\in \mathbb N, ...
0
votes
1answer
26 views

Fibonacci sequence property

I think its proof will be simple but. I dont know well When the difference of number of sequence in fibonnaci is 1 or 2, i know how to prove but this is not
2
votes
1answer
46 views

variant of Lagrange's four square theorem using a restricted set of squares

The well-known four square theorem states that any positive integer is the sum of at most four squares. Suppose that, instead of allowing all squares, I only consider the following set of squares: $$ ...
0
votes
1answer
17 views

Find the minimal polynomial of $\alpha=(1-\omega)/2$ over $\mathbb{Q}$ where $\omega$ is a primitive 8th root of unity

What I though about this question was that $(1-\omega)/2 = 1/2 -\omega/2$ so $\mathbb{Q}[\alpha] = \mathbb{Q}[\omega]$ over $\mathbb{Q}$ Thus its minimal polynomial of $\alpha$ over $\mathbb{Q}$ is ...
-1
votes
0answers
25 views

There are infinitely many odd numbers that are not expressible as the sum of a power of 2 and a prime

I have to prove this but so far I have struggled to find an example of an odd number that never gives a prime when powers of two are subtracted from it. Can someone give me some hints and/or examples ...
4
votes
2answers
380 views

Is PA the first axiomatization of arithmetic to be discovered? [closed]

Is Peano Arithmetic the first axiomatization of arithmetic to be discovered?
3
votes
0answers
43 views

Sum of integer squares with zero sum

This is something that I perhaps should know but don't. What is known about sums $\sum_{i=1}^k a_i^2$ subject to $\sum_{i=1}^k a_i=0$ where $a_i$ are integer? Specifically, which even integers can be ...
0
votes
0answers
23 views

Factorising into Gaussian primes $4+3i$

I want to find the Gaussian prime factors of $(4+3i)$ $$(4+3i)(4-3i) = 25=5^2$$ $$5: (2-i)(2+i)$$ so $$(4+3i)(4-3i) = 25=5^2 = (2-i)(2+i)(2-i)(2+i)$$ That was my answer but the solution says: ...
2
votes
2answers
56 views

If primitive root modulo $mn$, then primitive root modulo $m$ and $n$

Let $a$ be a primitive root modulo $mn$. Show that $a$ is also primitive root modulo $m$ and $n$. Showing $(a,mn)=1\Longrightarrow (a,m)=(a,n)=1$ is not a problem. The problem is showing $a^{\varphi ...
0
votes
0answers
18 views

Definition of $S$-ideles

This is a basic notational question. Let $K$ be a number field and $M_K$ the set of all places of $K$ with $S\subset M_K$ a finite subset. Write $\mathfrak J_K$ for the idele group of $K$ and ...
0
votes
1answer
30 views

If an integer $k$ is a divisor of an integer $n$, then $\frac{n}{k}$ is also a divisor of $n$?

A lot of these number theory ideas are popping up in my study of cyclic groups. In particular, in a note I came across. It is mentioned that: If an integer $k$ is a divisor of an integer $n$, ...
1
vote
1answer
56 views

Continued Fractions : Under which branch of mathematics do they come?

I wanted to know in which branch of Mathematics do Continued Fraction come? By branch I mean for example Geometry or Differential Equation are a branch of maths so is there any particular branch of ...
1
vote
3answers
53 views

How do I show that $\gcd(n,\frac{n}{k})=\frac{n}{k}$?

$\gcd(n,\frac{n}{k})=\frac{n}{k}$ Let $n, k$ be positive integers. This should be a trivial question but not having taken any classes in number theory I would like to be convinced with a simple ...
1
vote
1answer
24 views

Gcd of a Numbers

What is the Sum of GCD of this:where G is gcd of two numbers G(1,x)+G(2,x)+G(3,x)+..G(x,x)
0
votes
0answers
21 views

Prove that 4 is not a primitive root modulo n, for $ n \ge 2$ [duplicate]

I want to prove that 4 is not a primitive root modulo n, for $ n \ge 2$ I know how to prove it for any prime n, but I don't know how to prove it for numbers bigger than 2 which are not primes. ...
-1
votes
0answers
25 views

Unique base $10$ representations of real numbers

Prove that the base $10$ representation of any real integer value is unique. More generally prove that the base $n$ representation of any real integer value is unique. This seems obvious to me, ...