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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2
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1answer
82 views

Find all positive integers $(x,y,z)$ such that $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ is integer

As stated in title, I would like to find solution to this problem: Find all positive integers $(x,y,z)$ such that $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ is also integer. I need idea how to solve ...
2
votes
1answer
100 views

Diophantine Equation: $x^2 + 3y^2 = 11z^2$

I am having difficulty solving the following problem: Prove rigorously that there is no integer solution for the Diophantine Equation $x^2 + 3y^2 = 11z^2$ except when $x = y=z = 0$. ...
0
votes
1answer
158 views

Prove a system of simultaneous Diophantine equations has no solution.

I've been asked to show that the system of simultaneous Diophantine equations has no solutions: $3x+6y+z=3$ $12x+3y+2z=5$ I don't even know how to approach this problem, any help would be ...
2
votes
3answers
76 views

Diophantine Equation with 3 Variables

Find all solutions to $2x + 3y + 4z = 5$. I know how to do it with two variables, but I'm confused on how to start this with three variables.
0
votes
1answer
33 views

If $p,q \in \mathbb{P}$, $a \in \mathbb{N}$, $q| {a^p-1 \over a-1}$ and $p \not | a-1$, then $q \equiv 1 \pmod{p}$

The above is just a conjecture, there is a possibility it's wrong. The idea for it didn't come out of the blue, but I had to see some numerical evidence to see if it holds up, and none contradicted it....
-2
votes
2answers
2k views

Calculating (a / b) mod p

I am basically calculating $^nC_r\bmod p$ where $p$ is a prime..... For large values of $n$ and $r$... As we know $^nC_r$ $$ \frac{n!}{r!(n-r)!} $$ I did a little study of calculating it, but I always ...
1
vote
2answers
55 views

Homework gcd. Show that $gcd(a_1,a_2,a_3,…a_k) = gcd(gcd(a_1,a_2),a_3,…a_k)$

Help me with this please show that $gcd(a_1,a_2,a_3,...a_k) = gcd(gcd(a_1,a_2),a_3,...a_k)$ How can I start?
0
votes
0answers
100 views

Prime Reflections

How would you describe the following pattern?: For each primorial from 30 onward, there exists a pattern in the arrangement of the prime factors of the composite numbers which I call "the mirror ...
3
votes
1answer
109 views

A theorem of Szemeredi in Erdos's paper

In his paper "A survey of problems in combinatorial number theory", on page 110, Erdos writes: Graham conjectured: Let $1 \le a_1 < a_2 < \cdots < a_n$ be $n$ integers. Then $$ \max_{i,j} ...
-1
votes
1answer
32 views

maximal subtorus of a connected commutative algebraic linear group [closed]

I'm wondering the following: is the maximal subtorus of a connected commutative algebraic linear group over $k$ a) normal and closed b) defined over $k$ (for $k$ a field of characteristic zero, ...
0
votes
2answers
59 views

Proving rationality given

Trying to find a proof that $(x^n -1)^{1/n}$ is rational/irrational given $x$ is rational and $n>3$. I've tried searching online and in libraries. It's hard to find.
0
votes
0answers
51 views

The Real Part of an Imaginary Number

Can any real number be the real part of an imaginary number? Can a mathematical expression which is equivalent to any real number be used as the real part of an imaginary number? If a series ...
5
votes
0answers
81 views

Numbers Made From Concatenating Prime Factorizations

I came across the following curious problem while playing around with my calculator. Take any positive integer $n$; for this example we'll use $216$. Create a sequence as follows: Factor $n$ into ...
1
vote
1answer
53 views

Dirichlet characters - proof in a book

I found the following in a book and don't understand. Let $\chi$ denote a non-principal character modulo $q$ and $S(x)=\sum_{n\leq x}\chi (n)$. Then $\sum_{m>y} \frac{\chi(m)}{m} = \int_y^{\infty }...
18
votes
1answer
1k views

Can a double-factorial be a perfect square?

The title says it, basically. My question is $-$ for $ n \ge 2 $, can $n!!$ be a perfect square, where $!!$ represents the double-factorial? My conjecture is no, but I can't seem to be able to find a ...
1
vote
2answers
75 views

Solving congruence equations

Solve: $7x^6\equiv 11 \pmod{23}$ and $5^x\equiv 19 \pmod{23}$ I can solve simple congruence equations but how do I go about solving these?
3
votes
2answers
99 views

primitive root confusion

I found that 21 is a primitive root of 23. I wanted to find a primitive root of $529=23^2$ There is a theorem stating that if $x$ is a primitive root of $p$ and if $x^{p-1}$ is not congruent to 1 mod ...
0
votes
1answer
38 views

Combinations of sets raised to the power of a prime modulus

This is a problem out of the text Introduction to the Theory of Numbers by Niven, Zuckerman, and Montogmery and I am having quite a bit of trouble with it. I tried to prove it directly, but that didn'...
1
vote
0answers
47 views

On the partial zeta function

Let $F$ be a number field and $S$ be a finite set of places of $F$ including archimedian places. Let $\zeta^S(s)$ be the partial L-function, that is the meromorphic continuation of the product of ...
2
votes
2answers
100 views

Can Andrica's conjecture be proven by proving a tighter upper bound for prime gaps?

I checked some differences between square roots of various natural numbers and I am wondering what is required to prove Andrica's conjecture. Would a tighter upper bound for the prime gap above $n$ be ...
7
votes
0answers
113 views

Proving that $\left|\Re\left( \frac{1+i\sqrt{7}}{2}\right)^n\right| \to \infty$ [duplicate]

Let $u_n=\displaystyle\Re\left( \frac{1+i\sqrt{7}}{2}\right)^n$ Prove that $|u_n| \to \infty$ This appeared in a recent issue of French Revue de la Filière Mathématiques, as it was reportedly ...
0
votes
0answers
37 views

Is there a discrete relationship shared between patterns and series or sequences?

Maybe to clarify a little, I feel that patterns are related to sequences and series, in that a series or sequence can define a pattern. However I have yet to find any reference to such being the case ...
1
vote
1answer
60 views

Remainder of a summation divided by $2^{12}$

For a positive integer $n$, let $f(n)$ be equal to $n$ if there is an integer $x$ such that $x^2-n$ is divisible by $2^{12}$, and let $f(n)$ be $0$ otherwise. Determine the remainder when $$\sum_{n=0}^...
1
vote
1answer
19 views

$\binom{p^{\alpha}-1}{k} = (-1)^k\pmod{p}$? [duplicate]

I need to show that $$\binom{p^{\alpha}-1}{k} = (-1)^k\pmod{p}$$ for $0 \leq k \leq p^{\alpha}-1$. Not really sure how to start going about this... how should I transform the term on the left? ...
1
vote
1answer
597 views

Five digit numbers where each digit can appear up to three times

The question is to determine how many five-digit numbers there are (using the digits 0-9) where each digit can appear up to three times in the number. The total number of numbers that can be made ...
3
votes
1answer
153 views

Toy cryptographic hash function for education purposes?

I'm teaching some high school students about number theory and cryptography, and I'd like a hash function (or ideally, a family of hash functions) that I can use as simple demonstration for ...
1
vote
2answers
169 views

Prove that 12 has no primitive root

So I've got to prove that there exists no integer $a$ such that $a$ has order 4 mod 12. How can I do this? EDIT: Can I just try every integer less 12 and co-prime to 12 i.e. 5,7,11 Why does it ...
2
votes
2answers
276 views

How to prove 2 is a primitive root mod 37, without calculating all powers of 2 mod 37?

How can i prove 2 is a primitive root mod 37, without calculating all powers of 2 mod 37?
0
votes
1answer
39 views

Question about Linear Diophantine Equation.

A Linear Diophantine Equation is of the following form: $Ax+By+C=0$, where $\gcd(A,B)=d$ and $A=da,B=db$. If $(x_1,y_1)$ and $(x_2,y_2)$ are two solutions of the equation, then $b \textrm{ $|$ } (x_1−...
3
votes
0answers
81 views

Sum of roots (number theory)

Let k,m∈ℕ. Let a1,a2,...,ak>0 and b1,b2,...,bm>0. Let for all natural n, n>1 Prove that k=m. Prove that a1a2...ak=b1b2...bk Prove that if each of the two sets of numbers sort of growth, then ...
1
vote
4answers
78 views

$z^2=x^2+y^2$ Prove that $4\mid xyz$ ($xyz$ is divided by $4$)

$z^2=x^2+y^2$ where $x,\ y,\ z$ - integers Prove that $4\mid xyz$ ($xyz$ is divided by $4$) All possible rest in divided by $4$ in this case is $1$. That's all I noticed.
1
vote
1answer
84 views

Find all positive integer that $2^{2^n}+5 $ is a prime number. [duplicate]

Find all nonnegative integer that $2^{2^n}+5 $ is a prime number. For $n=0$ we have $7$ - correct For $n=1$ we have 9 - false For $n=2$ we have 21 - false For $n=3$ we have 259 ... Maybe any ideas ...
3
votes
0answers
49 views

Decide if radical expression equals a given rational number

Given a radical expression an a rational number. Is there an algorithm to decide if the expression equals the number? Example: $(\sqrt{\sqrt[5]{74} - \sqrt[14]{78}})^{356}+\sqrt[6]{63} \overset{?}= ...
3
votes
3answers
69 views

Finding all solutions for to the equation $x^3 = 0\ {\rm mod}\ 9$

How do I go about finding the solutions to: $$ x^3 = 0\mod 9 $$ Any help is greatly appreciated thank you
0
votes
1answer
40 views

A extended euclid algorithm related problem

A Linear Diophantine Equation is of the following form: Ax+By+C=0, where,gcd(A,B)=d and A=da,B=db.If (x1,y1) is a solution of the diophantine equation, every solution is of the form: x=x1+bt,y=y1−at ∀...
4
votes
1answer
60 views

Factorial equation

I'm trying to find all nonnegative integer solutions to $x!^2=z!$. Intuitively, I think the solutions are the trivial ones with $x=0,1$ and $z=0,1$. I'm not sure how to show that there is no more ...
0
votes
1answer
32 views

Prove for all $x \in \mathbb{R}$, there is some $y \in [0,1)$ such that $x \equiv y \mod \mathbb{Z}$

So my logic is as such choose any $x$ say $99.05$. Then I can find $y \in [0,1)$ such that $99.05-y \in \mathbb{Z}$ doesn't $y$ have to be $0.05$? Congruences are a little more difficult when you let $...
0
votes
1answer
56 views

Diophantus problem

I was given following problem as an example of early mathematics with the solutions. But it seems i can't understand from where they are getting the 35z^2 = 5 from in the solutions. Could someone ...
2
votes
0answers
74 views

The Existence of “Simple” Prime Generating Functions

Obviously, we do not know an explicit and easily manipulable formula for finding every prime - that is, a function $f(n)$ which yields the $n^{th}$ prime. I've seen plenty of formulas that "cheat" in ...
0
votes
2answers
37 views

How can I use the congruence property to determine GCD?

As per my text, the congruence property is: If a > 0, b, and b' are integers such that $$b \equiv b' (mod\ a)$$ then $$(a,b) = (a,b')$$ I'm trying to use that to determine (7,150) and (28,-288). Any ...
0
votes
1answer
58 views

Can we use the natural logarithm to find a previous prime?

Using any natural number $n \geq 3$ , can we set up a formula with the natural logarithm of something $x$ to find a previous prime? My calculations tell me strongly that the answer is yes, but I have ...
10
votes
1answer
240 views

Cube roots of five

This is not really homework. I might be able to do this myself in time, from the methods in Ireland and Rosen. Note that every number has exactly one cube root $\pmod q$ for any prime $q \equiv 2 \...
3
votes
1answer
46 views

Show $\sum\limits_{b=0}^{p-1}\left(\frac{b}{p}\right) = 0$

Show $\sum\limits_{b=0}^{p-1}\left(\frac{b}{p}\right) = 0$ The resources I have consulted said to use the fact that the number of quadratic residues $\text{mod } p$ is $\frac{p-1}{2}$ but I have no ...
0
votes
2answers
47 views

Decimals and Rational numbers

How do you prove: Q1) Why is every rational number (say m/n, where m and n are both positive integers) either a terminating or a repeating decimal? Q2) Why is every repeating decimal (or terminating ...
2
votes
1answer
50 views

smallest element order $p$ in $\mathbb Z^*_{p^2}$

I would like to write an efficient algorithm to find the smallest element of order $p$ in $\mathbb Z^*_{p^2}$, where $p$ is a prime number. Therefore I calculate $a^{p-1} \pmod{p^2}$ for every ...
0
votes
2answers
113 views

Solving a Linear Diophantine Equation

A Linear Diophantine Equation is of the following form: $Ax+By+C=0$, where $x_1 \leq x \leq x_2$ and $y_1 \leq y \...
2
votes
2answers
38 views

Summing powers of complex root of unities gives 0

I have a question regarding a proof. Let $z_N$ denote the complex N'th root of unity, from which we have the identities $(z_N)^n=1$ $\sum_{i=0}^{N-1}{(z_N)^i}=0$ Now let $N=r\cdot t$ and let $H^\...
5
votes
1answer
159 views

When is the next palindrome?

Okay, this is more just for fun than anything else. I'm driving in my car today, (true story) and my odometer is about to hit $81,818$. So, being a math nerd and all, I immediately see the pattern ...
2
votes
0answers
40 views

Does a random binary sequence almost always have a finite number of prime prefixes?

Does a random binary sequence almost always have a finite number of prime prefixes? Specifically, let $x = \sum_{1 \le i}{2^{-i} \cdot x_i}$ with $x_i \in \{0,1\}$ be a random real in $[0,1)$, $X_i = ...
4
votes
1answer
98 views

Some questions on the formation of the BSD conjecture

I'm quite curious how Birch and Swinnerton-Dyer formed their famous conjecture in the beginning of 1960s. I read some papers of Birch and Swinnerton-Dyer, as well as some papers of Tate and several ...