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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2
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1answer
72 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
31 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 ...
1
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3answers
42 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
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1answer
32 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 ...
-1
votes
1answer
32 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
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2answers
35 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
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0answers
59 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
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1answer
66 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} ...
-4
votes
1answer
185 views

Count good numbers in between L and R

Let length(A) denote the count of digits of a number A in its decimal representation. All non-negative numbers of length 1 are Good. Further, a number X with length(X) $≥ 1$ can also be considered ...
0
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0answers
28 views

String theory - working included [duplicate]

I'm really not too sure if I am correct or even on the right track with regards to the following question - any help is appreciated. Consider strings of five decimal digits. What are the number of ...
0
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1answer
21 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
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2answers
36 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
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0answers
40 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 ...
0
votes
0answers
20 views

Approximation of Spline Functions

Define a function g by the equation $$ g(x)= \begin{cases} 0, & \text{ when } t_0 \leq x \leq 0 \\ x, & \text{ when } 0 \leq x \leq t_n\ \end{cases} $$ Prove that every first-degree spline ...
0
votes
0answers
30 views

Proof Using Mathematical Induction that a set S equals Natural Set

I am trying to prove that if $S$ is subset of $\mathbb{N}$, such that a) $2^k \in S$ for all $k \in \mathbb{N}$. b) If $k \in S$ and $k \ge 2$, then $k-1 \in S$. Prove that $S = N$. I am trying ...
4
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0answers
42 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
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1answer
39 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 ...
17
votes
1answer
730 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 ...
2
votes
1answer
53 views

What is the maximum difference between two successive real numbers in the given floating point representation?

The following is a scheme for floating point number representation using 16 bits. Sign :- Bit 15 Exponent:-Bit 14-9 Mantissa :- Bit 8-0 Let $s, e,$ and $m$ be the numbers represented in binary in ...
3
votes
3answers
97 views

when ${\rm gcd} (a,b)=1$, what is ${\rm gcd} (a+b , a^2+b^2)$?

I want to prove above statement "what is ${\rm gcd} (a+b , a^2+b^2)$ when ${\rm gcd}(a,b) = 1$" I've seen some proofs of it, but i couldn't find useful one. here is one of the proof of it. some ...
1
vote
2answers
39 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?
2
votes
2answers
55 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
23 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 ...
1
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0answers
20 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
78 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 ...
6
votes
0answers
87 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 ...
0
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0answers
22 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
votes
0answers
35 views

How do you make the lighest change possible?

Suppose you have coin denominations $1 = c_1 < c_2 <... < c_k$ each with associated weight $w_1, ..., w_k$ and that you are trying to make change for $n$ cents. How can you make the ...
1
vote
1answer
45 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 ...
1
vote
1answer
17 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
30 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
28 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
0answers
73 views

Longest sequence of distinct squares which sum to a given number

I want to find the longest sequence of distinct integer squares $1 \le a_1^2 < a_2^2 < \dotsb < a_k^2$ such that $$a_1^2 + \dotsb + a_k^2 = n$$ where $n$ is a given [positive] integer. In ...
0
votes
1answer
37 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
52 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
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1answer
18 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{ $|$ } ...
3
votes
0answers
49 views

Are there any positive integers $a, b, c, d$ such that both $(a, b, c)$ and $(b, c, d)$ are Pythagorean triples?

Pythagorean triple is a triple of integers $(a, b, c)$ such that $a^2+b^2=c^2$. Is there any Pythagorean triple such that, not only $a^2+b^2$, but also $b^2+c^2$ is a square number? If not, how to ...
3
votes
0answers
72 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
57 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
54 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
31 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
54 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
18 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
45 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
30 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
32 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 ...
1
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0answers
30 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
27 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
34 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 ...
8
votes
1answer
165 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 ...