2
votes
1answer
40 views

Twin Prime conjecture current status

Can someone help me with a link to read about the status of the Twin Prime conjecture. I have browse on the internet and have read some articles but still I have no clue of the updated status of Twin ...
0
votes
3answers
10 views

Solving a system of linear congruences with a small mistake

I having trouble solving the system of linear congruences: $x \equiv 1 (mod 2)$ (a) $x \equiv 1 (mod 3)$ (b) so what i do is from (a) $x = 2y + 1$ and into (b) $2y + 1 \equiv 1 (mod 3)$ so $2y ...
1
vote
4answers
70 views

Number theory with positive integer $n$ question

If $n$ is a positive integer, what is the smallest value of $n$ such that $$(n+20)+(n+21)+(n+22)+ ... + (n+100)$$ is a perfect square? I don't even now how to start answering this question.
1
vote
2answers
51 views

Proof that 2 and 3 are the only siamese twins that exist!

Let us say that two prime number p and q are siamese twins if |p-q|=1. List all the siamese twins that exist, and prove your list is complete. Proof: 2 and 3 are prime numbers and 3-2=1. Therefore 2 ...
-4
votes
2answers
46 views

I need help with a relatively prime proof! [closed]

I would like to understand this completely Prove that if n is an odd integer, then gcd(n,n+2)=1
1
vote
0answers
66 views

Is Modeling the Future of Mathematics [closed]

So my Differential Equations professor today said one sentence then left the room. It was the weirdest moment of my college career but I felt like he really tried to say something. He walked in 1 ...
0
votes
2answers
36 views

Different values of $x$ and $y$ between $\sqrt{39}$ and $\sqrt{224}$

If $x$ and $y$ are whole numbers between $\sqrt{39}$ and $\sqrt{224}$, then how many different values can $x$ + $y$ have? OK, first I found that the set numbers are: $$7, 8 ,9 ,10 ,11 ,12, 13,14$$ ...
-1
votes
5answers
99 views

Is this a true theorem? [closed]

I'm trying to prove the existence of the following theorem: If $n,p \in \mathbb{N}$, then $(p+1)^n = 1 \mod p$ Is this theorem true? I think it is, but I don't know how to prove it! Thanks!
1
vote
4answers
40 views

How to find $k$ in $6^{k} \equiv 2 \mod {13}$

Find for which $k$ is $6^{k} \equiv 2 \mod {13}$ I'm having trouble with these types of question in my cryptography class. This is part of Diffie–Hellman algorithm for calculating a shared key. ...
0
votes
1answer
21 views

Need help proving cardinal of $\{n \in \mathbb{N}: n \le x, d|n\}= \lfloor \frac{x}{d} \rfloor$

I need to show this $\{n \in \mathbb{N}: n \le x, d|n\} = \lfloor {\frac{x}{d}} \rfloor$ but I don't know where to start =(
0
votes
0answers
28 views

Question on perfect numbers: $\sigma (n) = n \rightarrow \sigma (kn) = kn$? [duplicate]

I'd like to proof that if $\sigma (n) = n$, where $\sigma(n)$ is the sum of all divisors of n$ then follows $\sigma (kn) = kn, k \not= 1$. I have $\sigma (kn) = \sum_{x | kn} x = ...??$
0
votes
0answers
34 views

number theory expressing as notation [on hold]

Surf the internet and find a theorem of number theory. State the claim of the theorem, and then express it in logical notation. (Example: It is a theorem of number theory that if n is an even natural ...
1
vote
2answers
21 views

Given a congruence equation ax = b (mod n), how can I prove this GCD?

I am given the equation $ax = b (mod$ $n)$ and that $d = (a,n)$. Suppose that $x_o$ is a solution to the equation. I need to prove that d is the greatest common divisor of not only a and n, but b as ...
0
votes
4answers
39 views

What modular arithmetic theorem is being ignored here?

Suppose $4x\equiv 6 \pmod {18} $ Then $2x\equiv 3 \pmod 9$ Then $6x\equiv 9 \pmod 9$ Then $6x\equiv 0 \pmod 9$ Then $x\equiv 0 \pmod 9$ Then $x=9k$ vs. Suppose $4x\equiv 6 \pmod {18}$ Then ...
1
vote
2answers
27 views

Can someone help me solve this system of congruences?

I'm a little new to congruences but I think I have it right. I started with the following congruences: $19x \equiv 5 \pmod{2}$ $19x \equiv 5 \pmod{3}$ $19x \equiv 5 \pmod{5}$ $19x \equiv 5 ...
0
votes
1answer
34 views

Alternative proof of Liouville's approximation theorem

Let $\alpha\in\mathbb{R}$ be an algebraic number of degree $d\geq2$. I am asked to prove Liouville's approximation theorem using the fact that $$ \mathop{\text{den}}(\alpha)^d ...
4
votes
4answers
105 views

Show $\left( n!\right)^2 > n^n$.

If $n > 2$, show that $$\left(n!\right)^2 > n^n$$ Although the problem is pretty obvious, I couldn't come up with a rigorous proof. I was thinking some sort of AM-GM, but couldn't build ...
1
vote
3answers
74 views

Find the smallest k.

Find the smallest $k$, $$\sum_{n = 0}^{1013} \binom{2n}{n}k^n \mod{2027} \equiv 0$$ The problem was posted on Brilliant, but no one has submitted a solution yet. I tried expanding to get an idea, ...
1
vote
2answers
36 views

Is this a good expression of the claim using logical notation?

Four Square Theorem: Every positive integer can be written as a sum of four integer squares. Expressed in logical notation : $$\forall n>0 =a_{0}^{2} + a_{1}^{2} + a_{2}^{2} + a_{3}^{2}$$
0
votes
1answer
68 views

Problem of Integer Pairs

Prove that there exists infinitely many pairs of positive integers $(m,n)$ satisfying the following properties: $\gcd(m,n)=1.$ $(x+m)^3=nx$ has three distinct integer solutions.
0
votes
2answers
21 views

How can I prove this relationship between primes and congruences?

Suppose $p$ is prime and $x^2\equiv 1 (\bmod~p)$. Prove that $x\equiv \pm1 (\bmod~p)$. To start, does the statement in the proof imply that $p|(x+1)$ and $p|(x-1)$ or is it an "or" relation. if it is ...
0
votes
3answers
51 views

No primitive root modulo $2^n$ for $n\ge 3$

Prove that there is no primitive root modulo $2^n$. I'm not sure how to begin proving this. I know $\varphi(2^n)=2^{n-1}$, thus a primitive root $a\in\left(\dfrac{\mathbb{Z}}{2^n\mathbb{Z}}\right)^*$ ...
3
votes
4answers
68 views

Can anyone prove for every $a,b \in \mathbb Z^+ < p$ ( $p$ is a prime), $p \nmid ab$?

Can anyone prove for every $a,b \in \mathbb Z^+ < p$ ( $p$ is a prime), $p \nmid ab$? I was trying my best to do the problem but like I don't know where to start or anything!
0
votes
0answers
59 views

Heavy Application of Fermat's Theorem

Show that if p = 4k + 3 is a prime, then the product of all the even integers less than p is congruent modulo p to either 1 or -1. I know now that Fermat's theorem implies that 2^((p-1)/(2)) == 1 or ...
1
vote
6answers
856 views

Is this a theorem in Number Theory? I can't find this in my textbook

"If $a \equiv b \pmod m$ and $a \equiv b \pmod n$ and $gcd(m,n)=1$, then $a \equiv b \pmod {mn}$ " Is that a true theorem? I can't find it in my textbook!
2
votes
3answers
120 views

Finding the values of $A,B,C,D,E,F,G,H,J$

Given that the letters $A,B,C,D,E,F,G,H,J$ represents a distinct number from $1$ to $9$ each and $$\frac{A}{J}\left((B+C)^{D-E} - F^{GH}\right) = 10$$ $$C = B + 1$$ $$H = G + 3$$ find (edit: ...
0
votes
4answers
36 views

show that $ϕ(n/d)=g(d)$ where $ g(d)=\# \{a \in \{1,\dots ,n\} | GCD(a,n)=d\}$

This is a homework question and I am to show that $$ϕ(n/d)=g(d)$$ where $g(d)$ is the number of $\{a \in \{1,\dots,n\}|\, GCD(a,n)=d\}$ I have justified earlier that the sum over all divisors $d$ of ...
0
votes
2answers
29 views

Proof $ GCD(a,b) = GCD(a, b-a) = GCD (a, r_b) $

let $a,b \in \mathbb{N}$ and a < b. let $r_b$ the rest when dividing b through a. (1) If $r_b$ is the rest, then there exists a q so that $ b = q*a + r_b $. (2) Now I show: $gcd(a,b) = gcd(a, ...
0
votes
2answers
34 views

Proof for: GCD and divisibility

I need some hints how to proof something like the following: Let $a,b \in \mathbb{Z}$ with $a,b \not= 0$ and let $\gcd(a,b)=d$. (1) For any $m,n\in \mathbb{Z}$ we have $d \mid ma+nb$. (2) There ...
5
votes
1answer
101 views

Riemann Zeta Function Non-Vanishing on the Line $\mathrm{Re} \; z = 1$

The result quoted in the title is usually a stepping stone in the proof of the prime number theorem and I am familiar with the usual argument for this result. The other day my professor was telling ...
1
vote
0answers
64 views

The von Mangoldt Function and the Prime Number Theorem

Right now I am stuck trying to show the following: Show that $\sum_{n \leq x} \frac{\Lambda(n)}{n} - x$ tends to a finite limit if and only if the prime number theorem is true. If someone could ...
1
vote
0answers
26 views

Trouble Identifying a “Psi” Function in Number Theory

In these lecture notes on number theory I am reading I came across the notation $\Psi(e^t;a,q)$ in connection with the Dirichlet theorem on arithmetic progression. I was hoping someone could help me ...
0
votes
0answers
37 views

Module in $\mathbb{Z}$ and characteristics

I'd like to show some basic characteristics for a module in $\mathbb{Z}$. The module in Z has been defined as: "A subset $\not0 \not = M \subseteq \mathbb{Z}$ is called module, if M is closed under ...
2
votes
1answer
43 views

Continuation of the Riemann Zeta Function

I am actually aware of the argument showing $\zeta$ has a meromorphic extension to $\mathbb{C}$ with a single pole at $z = 1$. On a recent number theory exam, however, one of the questions asked to ...
1
vote
0answers
55 views

EXERCISE 2.7.2 fron Alon and Spencer probabilistic method.

Prove that there is a positive constant $c$ so that every set $A$ of $n$ nonzero reals contains a subset $B\subset A$ of size $|B| > cn$ so that there are no $b_{1},b_{2},b_{3},b_{4}\in B$ ...
-1
votes
2answers
62 views

how to find nth term of different fibonacci series with golden ratio [duplicate]

what i know : if i want to find $Nth$ term of a fibonacci series like : 1 1 2 3 5 8 13 21 ....... then to find $6th$ term we use golden ratio ...
1
vote
2answers
80 views

combinatorics and divisibilty

in how many ways we can form a $8$ digit numbers from $1,2,3,4,5$ with repetition allowed & divisible by $8$. MY APPROACH : to be divisible by 8 : last 3 digit of the no. must be divisible by 8 ...
2
votes
1answer
34 views

Equidistribution of $\{\xi_n\}$ where $\xi_n = <n\frac{p}{q}>$ $p,q$ rel. prime

I'm working from Stein's An Introduction to Fourier Analysis, and there's a question (chapter 4 number 6): Let $\theta = \frac{p}{q} \in \mathbb{Q}$ where $\operatorname{gcd}(p,q) = 1$. Assume ...
0
votes
1answer
20 views

Condition for L and R to make the below scenario true?

You are a given a number N. N <= 10 ^ 9 . You are given a range of numbers L and R . ...
3
votes
3answers
55 views

Elliptic curve over $\mathbb{Q}$ cannot have $\mathbb{Z}_4\times\mathbb{Z}_4$ as a subgroup

Show that an elliptic curve over $\mathbb{Q}$ cannot have $\mathbb{Z}_4\times\mathbb{Z}_4$ as a subgroup. We've been told that for this problem, we are not allowed to use Mazur's Theorem. ...
3
votes
0answers
35 views

Number of points over elliptic curve is p+1 given…

Suppose that -1 is not a square in $\mathbb{Z_p}$. Show that the number of points on the elliptic curve $y^2=x^3+ax$ over $\mathbb{Z_p}$ is $p+1$. Hint: Use the fact that $x^3+ax$ is an odd function. ...
1
vote
3answers
94 views

Is -1 less than 0.1?

In a High School Maths Test, I presumed that since -1 has as much mathematical mass as a whole unit [-1 x -1 = 1, 1 x 1 = 1] and 0.1 represents one tenth of a unit, that -1 is greater than 0.1 -1 is ...
3
votes
3answers
37 views

Divisibility in base $7$ problem

Find all integers between $0 \leq a \leq 2400$ such they are divisible by $8$ and that their base 7 development has at least $3$ equal digits.
0
votes
0answers
31 views

observation on poulet numbers need generalizations

Let us take two different poulet numbers with common prime factor. For example, CASE-1: $p_1$ $= 341 = (11)(31)$ and $p_2$ $= 4681 = (151)(31)$, as $31$ is common prime factor. let us define some ...
2
votes
1answer
54 views

Natural number to Rational number and at the end POULET number

Generalize / prove or disprove the following statement. For any prime $p$ $>5$ and prime $q$, we get infinite natural numbers $N$ such that $N$$= (q-1)/(p-1)$. If $N$ is rational, instead of ...
1
vote
1answer
61 views

Geometrical series with 9

You have this infinite sum: $\frac{1}{9} + \frac{1}{99} + \frac{1}{999} + \frac{1}{9999} + ...$ Take a truncated sum (just $n$ terms) and consider the numbers on the right side of the point. Which ...
0
votes
2answers
44 views

What are all the values of $ x \equiv a^\frac{p-1}{2} \pmod p$

Suppose p is an odd prime and a $\in$ $\mathbb{Z}$ such that $ a \not\equiv 0 \pmod p$. What are all the values of $ x \equiv a^\frac{p-1}{2} \pmod p$ ? This is what I got so far: $ x^2 \equiv ...
-1
votes
1answer
47 views

Prove a property of the greatest common divisor of several numbers [closed]

Show that the greatest common divisor of the integers $a_1, a_2, ..., a_n$, not all $0$, is the least positive integer that is a linear combination of $a_1, a_2, ..., a_n$.
0
votes
0answers
104 views

$\Bbb Z[\sqrt{-2}]$ prime factorisation

$S:= \{ a + b \sqrt{-2} | a,b \in \mathbb{Z} \}$ We say a number is S-prime if it has no non-trivial factorisation. $\text{iv})$ Factorise $171$ into primes in $\Bbb Z$ and into S-primes in $S$ I ...
1
vote
2answers
63 views

From $y^2=x^3+Ax^2+Bx$ to $y^2+(1-c)xy-by=x^3-bx^2$

I have two question How can I transfer with a change of coordinates from $$y^2=x^3+Ax^2+Bx$$ to $$y^2+(1-c)xy-by=x^3-bx^2?$$ In a note of Prof. Lozano "Elliptic Curves, Modular Forms and their ...