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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3
votes
1answer
67 views

Determine Units of a Ring $\mathbb{Z}[\alpha]$

I am trying to determine the units of $Z[\alpha]$ where $\alpha$ satisfies the monic polynomial $\alpha^4+\alpha^3+\alpha^2+\alpha+1$. I found $Z[\alpha] := \lbrace a+b\alpha+c\alpha^2+d\alpha^3\;|\; ...
0
votes
2answers
38 views

Show if (m,n) = 1, then for any # p, we have (p,mn) = (p,m)(p,n).

Show that if $(m,n) = 1$, then for any number p, we have $(p,mn) = (p,m)(p,n).$
7
votes
5answers
665 views

Induction hypothesis misunderstanding and the fundamental theorem of arithmetic.

The fundamental theorem of arithmetic is made of two parts: The existence part: There exist primes such that for any natural number $j$, we can write $j$ as a product of prime numbers. The ...
0
votes
0answers
57 views

2 player team knowing maximum moves

Given a list of N players who are to play a game. Each of them are either well versed in a move or they are not. Find out the maximum number of moves a 2-player team can know. And also find out how ...
5
votes
2answers
128 views

Infinitely many prime $n$ for which $n^2 = p + 8$ for some prime $p$.

How to prove that there exist an infinite number of prime $n$ for which $n^2=p+8$ for some prime $p$? Verification of the form $n^2=p+8$ where $n$ and $p$ are some $p$. $$\begin{array}{|c|c|} \hline ...
1
vote
1answer
48 views

Signed determinant of quadratic forms over Q_p

Let $W(k)$ be the Witt-Ring of the field $k$. in this script http://math.uga.edu/~pete/quadraticforms2.pdf at the bottom of page 2 the signed determinant is introduced by $d^\pm (q) = ...
2
votes
0answers
58 views

Birthday problem & primes

Let $\pi_k(n)$ be the almost prime counting function, then $\pi_k(2^kn)$ reaches a max value, since $\pi_k(2^kn)=\pi_{k+1}(2^{k+1}n)$ for large enough $k$. (eg, ...
2
votes
1answer
68 views

$\text{lcm}(1,2,3,\ldots,n)\geq 2^n$ for $n\geq 7$

I can prove that $\text{lcm}(1,2,3,\ldots,n)\geq 2^{n-1}$. Newly, i read in a paper that for $n\geq 7$ we have: $$\text{lcm}(1,2,3,\ldots,n)\geq 2^n$$ Can you prove it? (this inequality is an ...
1
vote
1answer
54 views

If $a+\frac{1}{b}, b+\frac{1}{c}, c+\frac{1}{a}\in\mathbb{Z}$, find $a+b+c$. [duplicate]

Let $a,b,c$ be positive rational numbers such that $a+\frac{1}{b}, b+\frac{1}{c}, c+\frac{1}{a}$ are all integers. Find all the possible values of $a+b+c$. it would be too complicate to solve by ...
0
votes
2answers
41 views

$A^7 \not\equiv A(\mod 13) \Rightarrow A^{78} + 1 \equiv 0 (\mod 169)$

Let variable $A$ is integer and $A^7 \not\equiv A(\mod 13)$. Prove that $A^{78} + 1 \equiv 0 (\mod 169)$ Could someone explain, how to solve this type of problems? Any help would be greatly ...
4
votes
3answers
124 views

Interesting behavior of $\frac{n}{v_2(n!)+1}$.

I've lately noticed some interesting behavior from the values of the function $f(n)=\frac{n}{v_2(n!)+1}$, Where $v_p(n)$ is the $p$-adic valuation of $n$, and we also know that ...
2
votes
0answers
81 views

Questions about central polygonal numbers $1, 2, 4, 7, 11, 16, 22, 29, 37, 46,\cdots$

Formula for Central polygonal numbers is $\frac{n(n+1)}{2} + 1$, if $n=1$ or $n$ is prime, we get the new sequence $A$: 2, 4, 7, 16, 29, 67, 92, 154, 191, ... It seems that all primes either is ...
0
votes
2answers
355 views

Finding Coprime triplets

Given a sequence a1, a2, ..., aN. Count the number of triples (i, j, k) such that 1 ≤ i < j < k ≤ N and GCD(ai, aj, ak) = 1. Here GCD stands for the Greatest Common Divisor. Example : Let N=4 ...
0
votes
1answer
51 views

Counting solutions mod p of a polynomial equation

Hello: Does somebody know if the following is true?: Let $f\in \mathbb{Z}[X]$ be a monic irreducible polynomial of degree $n$. Then there exists a positive integer $N$ and ...
1
vote
3answers
80 views

Find all values that make the expression a perfect cube [closed]

Find all the positive integers $n$ such that $n^3-n$ is a perfect cube.
4
votes
1answer
26 views

Finding the smallest $x$ such that $ax\equiv b\mod m$

I'm looking to solve for $x$ in the equation $ax\equiv b\mod m$ and I wish to find the smallest $x$ which satisfies this. How would I go about doing this, in the general case? (This is for a ...
5
votes
1answer
79 views

Binomial Congruence

How can we show that $\dbinom{pm}{pn}\equiv\dbinom{m}{n}\pmod {p^3}$ for positive integers m and n and p a prime greater than 5? I can do it for mod p^2 but Im stuck here.
3
votes
1answer
42 views

Can a Mersenne number be a power (with exponent > 1) of a prime?

Let $n \geq 1$ and consider the (Mersenne) number $M_n = 2^n-1$. Is it possible that $M_n = p^k$ for some prime $p$ and some (necessarily odd) $k > 1$? Thanks in advance.
0
votes
1answer
25 views

Quaternary Quadratic Forms

What is a Quaternary Quadratic Form? I've looked for a definition online and cannot find a precise clear definition. I am not taking a course. Just reading about quadratic forms. Thank you.
1
vote
3answers
98 views

Is there an algorithm help us to write each even number as sum of primes numbers? [closed]

1) Is there an algorithm help us to write each even number as sum of primes numbers : for example :$4=2+2$ $8=3+5$ where : $3,5,2$ are primes 2) why we couldn't writing all odd numbers as sum of ...
0
votes
0answers
14 views

Logarithm of the basic Lubin-Tate formal group

Let $K$ be a local field with finite residue field of cardinality $q$. Let $\pi$ be a uniformizer. The basic Lubin-Tate group (associated to $\pi$) is the unique formal group associated to the ...
0
votes
0answers
34 views

Riemann Zeta Function at Real Values of the variable s

My question is: Is the Riemann Zeta function for real values of $s$ $( s = \sigma + 0\,i)$ a monotone function of $\sigma \,$?
2
votes
0answers
37 views

Tau Summatory Function

It is well known that the divisor summatory function can be calculated in $O(x^{1/2})$ via $$D(x)=\sum_{n\le x} d(n) = 2 \sum_{k=1}^{\lfloor \sqrt{x}\rfloor} \lfloor\frac{x}{k}\rfloor - \lfloor ...
0
votes
0answers
17 views

to maximize the summation

let F=$∑i=1$ to $N$ $((abs(A[i]-X))^P mod $K$)mod K$ $A[1..N]$ is an array with $N$ elements, the problem is to find $X$ such that the above summation F maximized where $X$ can take any value from ...
3
votes
0answers
33 views

Prime Triangle:: How to find the position(row and column) of prime number in a triangular arrangement

I was working on problem which asks the position of a prime number in a triangular arrangement. If we arrange the all prime up to 10^8 as shown in image we can find the row and column number of a ...
3
votes
1answer
52 views

A finite sequence of natural numbers, whose sum equal its product:

The following pattern yields a finite sequence of natural numbers, whose sum equal its product: $A_1=k$ $A_2=2$ $A_3,\dots,A_k=1$ A few examples: $A_n=2,2$ $A_n=3,2,1$ $A_n=4,2,1,1$ ...
13
votes
3answers
553 views

Are there numbers that describe themselves in some base but not according to the pattern 6210001000?

Call the first digit of a number digit 0. The digit after that digit 1, and so on and so forth. In base 10, the number 6210001000 describes itself, because digit 0 is 6 and it has 6 0s. Digit 1 is 2 ...
4
votes
1answer
39 views

q-expansion principle for modular forms

Let $f(z)$ be a modular form of some integral weight $k \geq 0$ and level $\Gamma_1(N)$ (I insist I want $\Gamma_1(N)$, not $\Gamma_0(N)$ or $\Gamma(N)$). Thus for any $d \in (\mathbb Z/N\mathbb ...
1
vote
3answers
83 views

Find all distinct integers $(x, y):x\log y = y\log x$?

Find all distinct integers $x$ and $y$ that satisfy the following equation. $$ x\log y=y\log x. $$ Obviously, if $x=y$, the equation is satisfied. I found $x=2$, and $y=4$. I think we cannot find ...
26
votes
4answers
493 views

Geometric intuition for $\pi /4 = 1 - 1/3 + 1/5 - \cdots$?

Following reading this great post : What are some interesting cases of $\pi$ appearing in situations that are not / do not seem geometric?, Vadim's answer reminded me of something an analysis ...
0
votes
1answer
45 views

SMO 2013 Junior First Round Q35. [duplicate]

In a competition that i have recently taken part in, one of this questions popped out: $2^{29}$ has nine distinct digits. Find the digit that is not in the sequence. My answer will be placed as a ...
3
votes
1answer
59 views

Improving Bertrand's postulate

Recall that Bertrand's postulate states that for $n \ge 2$ there always exists a prime between $n$ and $2n$. Bertrand's postulate was proved by Chebyshev. Recall also that the harmonic series $$ 1 + ...
0
votes
0answers
27 views

Hardness of bounded modular square root of 1

If we know any square root of 1 modulo N different from 1 and N-1, then we can find a nontrivial factor of N. So to find such a square root has a certain hardness. In fact, if in general we ask to ...
1
vote
1answer
32 views

Issue in the first French edition of Serre's local fields

I've been reading Serre's Corps Locaux, and I believe my copy is a first edition, as there's only one copyright date listed, 1968. I believe I found an issue on page 57, which (if you're looking at ...
11
votes
5answers
1k views

How to tell if a Fibonacci number has an even or odd index

Given only $F_n$, that is the $n$th term of the Fibonacci sequence, how can you tell if $n \equiv 1 \mod 2$ or $n \equiv 0 \mod 2$? I know you can use the Pisano period, however if $n \equiv 1$ or ...
1
vote
2answers
50 views

To find the right most non zero digit

When expanded 30! ends in 7 zeroes. Find the first non zero digit from right?
0
votes
0answers
35 views

another series problem [duplicate]

I need help in simplification below are the two formulas for AGP series: if $n$ is even $a\cdot r^{(n-1)/2} + d\cdot( 1 + r + r^2 + r^{(n-1)/2})$ if $n$ is odd $a\cdot r^{(n-1)/2} + d\cdot( 1 + ...
0
votes
0answers
40 views

need help in simplification

I need help in simplification below are the two formulas for AGP series: if $n$ is even $a\cdot r^{(n-1)/2} + d\cdot( 1 + r + r^2 + r^{(n-1)/2})$ if $n$ is odd $a\cdot r^{(n-1)/2} + d\cdot( 1 + ...
1
vote
1answer
34 views

Reverse Modulus Operator with given condition

I have an equation: $$ x^2 \mod p = z $$ $p$ and $z$ are given. $x$, $p$ and $z$ are positive integers and a maximal value of $x$ is given (say $M$). $p$ is a prime. How can i calculate (multiple ...
5
votes
4answers
128 views

When is the sum of two squares the sum of two cubes

When does $a^2+b^2 = c^3 +d^3$ for all integer values $(a, b, c, d) \ge 0$. I believe this only happens when: $a^2 = c^3 = e^6$ and $b^2 = d^3 = f^6$. With the following exception: $1^3+2^3 = 3^2 + ...
6
votes
1answer
46 views

Sequences where each number is a divisor of one less than the next

Let $N,k$ be fixed. Call a sequence of positive integers $a_1,a_2,\dots,a_k$ good if for each $i$, $a_i$ is a divisor of $a_{i-1}-1$. Consider the set $$S = \{a_i : a_1,\dots,a_k \text{ is a good ...
1
vote
2answers
39 views

How are the algebraic and Constructible numbers enumerable?

I was reading http://en.wikipedia.org/wiki/Algebraic_number which atates that the algebraic numbers are countable. if i am correct this means that there are "only" $\aleph_0 $ of them and that they ...
2
votes
3answers
96 views

Proof that a prime can't divide a multiplication of two reminders of it

Let $p$ be a prime number. Let $r_1$, $r_2$ be integers such that $1\leq r_1 < p$ and $1 \leq r_2 < p$. How to prove that $p \nmid r_1r_2$? I know one way to prove this. It can be proved by ...
2
votes
0answers
86 views

Find the limit of $\sum \frac{1}{log^n(n)}$

Working on convergence and divergence of infinite series, I recently focused my attention on the summation $$\displaystyle\sum\limits_{n=2}^{\infty} \frac{1}{log^n(n)}$$ While proving the convergence ...
0
votes
3answers
79 views

Largest possible value of consecutive integers

Find the largest possible value of $k$ for which $3^{11}$ is expressible as the sum of $k$ consecutive positive integers?
0
votes
1answer
42 views

Number theory - simple proof on proving an integer

Let n,x be a Positive Integer. Prove xn is an integer This seems very simple. How do I prove this?
1
vote
1answer
30 views

remainder of positive integer

An integer greater than 1, when divided by an integer say k, (2<=k<=11) leaves a remainder 1. find the difference of such two least integers.?
3
votes
2answers
30 views

Divisor problems when it is in the high power format

Find the number of positive integers which exactly divide 10^999 but not 10^998?
0
votes
1answer
21 views

Number System Problems

Let p,q,r,s,t be consecutive positive integers such that q+r+s is a perfect square and p+q+r+s+t is a perfect cube. Find the smallest possible value of r?
0
votes
4answers
72 views

Find all the integer solutions [closed]

Let $p,q$ be prime numbers, find all the integer solutions to: $$q^2(p-1) = (p+1)(q+1)$$