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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0answers
12 views

Calculating a floor sum

Is there any explicit closed form expression for $\sum_{k=1}^{\dfrac{p-1}2} \bigg\lfloor \dfrac{kq}p \bigg\rfloor-\bigg\lfloor \dfrac{k(q-1)}{(p-1)} \bigg\rfloor$ , where $p,q$ are odd primes ?
-34
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8answers
3k views
+100

Unique Representation and The Fundamental Theorem of Arithmetic

While reading this thread Is 1 a prime number?, I recalled that The Fundamental Theorem of Arithmetic (FTA) which says that every positive integer greater than 1 can get written uniquely as a product ...
0
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1answer
28 views

divisibility on prime and expression

This site is amazing and got good answer. This is my last one. If $4|(p-3)$ for some prime $p$, then $p|(x^2-2x+4)$. can you justify my statement? High regards to one and all.
19
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5answers
2k views

The myth of no prime formula?

Terence Tao claims: For instance, we have an exact formula for the $n$th square number - it is $n^2$ - but we do not have an exact formula for the $n$th prime number $p_n$! "God may not play ...
11
votes
2answers
701 views

Are primes randomly distributed?

There is a famous citation that says "It is evident that the primes are randomly distributed but, unfortunately, we don't know what 'random' means." R. C. Vaughan (February 1990) I have this very ...
-4
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1answer
74 views

Outline approach to Collatz 3n+1 conjecture / Criticism needed

This is a sketch of an approach to proving the Collatz 3n+1 conjecture true along the following lines. Instead of trying to show there are no loops and no sequences that increase without bounds, ...
-4
votes
0answers
32 views

Identifying Symbols [on hold]

When you see $x$ written on a piece of paper you automatically identify it. When you yourself write $x^2 + 2x = 0$ The $x$ you write in $x^2$ differs from the $x$ you write in $2x$ just by a ...
3
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0answers
24 views

how to find the last non-zero digit of $n$

I want to know how to find the last non-zero digit of $n$. For example $n = 100!$ my try: First i have to know how much Zeros $100!$ has so i did this: $$E_{5}10 = \sum _{1\leq k <n} ...
12
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1answer
394 views

Fibonacci, compositions, history

There are three basic families of restricted compositions (ordered partitions) that are enumerated by the Fibonacci numbers (with offsets): a) compositions with parts from the set {1,2} (e.g., 2+2 = ...
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vote
2answers
45 views

Proving the general formula [nx] where [.] is the floor function.

I've been trying to solve a exercise that asks me to prove the following generalization for the floor function: $$\lfloor nx\rfloor = \sum_{k=0}^{n-1} {(x + \frac kn)}$$ I've already proven the ...
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0answers
24 views

Simple number theory question involving division

I have a simple number theory question that involves a proof. Here is the question: If p divides abc and p does not divide a and p does not divide b prove that p does not divide ab. I'm sure this ...
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2answers
46 views

Are there particular techniques to find the general formula for an arithmetic function, neither multiplicative nor additive?

I was reading about the Euler phi function and the sigma function when I began to wonder how on earth one gets to the general formula for an arithmetic function. I'm not considering trivial formulae ...
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1answer
97 views

Do Prime Numbers have a Structure or do they sprout out Randomly among positive Integers? [duplicate]

Since the Order of Sequence of the Prime Numbers has not been found, it seems that all famous Mathematicians have opted for the random appearance of Primes.
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1answer
36 views

$P(n)$ is product of all digits of $n$. Find all $n$ such that $P(n)$ = $n^2−10n−22$.

$P(n)$ is defined as product of all digits of $n$ (decimal representation). Find all $n$ such that $P(n)$ = $n^2−10n−22$. I know the answer, which I will post later on in few days, but I want ...
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0answers
48 views

Count ways to reach Nth row

Given a N*M grid I need to reach last row with following operations : ...
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0answers
43 views

Door game between alice and bob

Alice and Bob are taking a walk in the Land Of Doors which is a magical place having a series of N adjacent doors that are either open or close. After a while they get bored and decide to do ...
6
votes
2answers
160 views

When $x^2+6xy+y^2$ a square number?

Find all natural numbers $x$ and $y$ such that $x^2+6xy+y^2$ is a square number. For example, $(x,y)=(2,3)$ or $(x,y)=(3,10)$. Obviously, we can consider $gcd(x,y)=1$.
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votes
1answer
345 views

Definition of a peculiar quotient group of isometries

I just wrote a text file to sum up my ideas about the Riemann Hypothesis. This text is a draft, and I don't expect people here to say if this approach is interesting or not (but if by chance you think ...
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0answers
35 views

Finishing conclusion of GCD proof?

I'm trying to prove that $a$ divides $bc$ if and only if $$\frac{a}{\gcd(a, b)} \mid c$$ I go in the right direction first (i.e. if $a$ divides $bc$ then $\frac{a}{\gcd(a, b)} \mid c$): We want to ...
0
votes
4answers
89 views

How can we find the smallest number $n$ such that $2^{2^n} + 1$ is not a prime.

How can we find the smallest Fermat number (i.e. in the form $2^{2^n} + 1, n \in \mathbb N$) that is not prime and show that it is indeed not a prime? Yes, when $n=5$, it is not a prime. How can we ...
1
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1answer
64 views

Does Zhang's result on primes makes RSA weaker?

I read from Finnish newspaper ( http://www.uusisuomi.fi/tiede-ja-ymparisto/72212-matemaattinen-ongelma-eli-2-300-vuotta-mies-subway-tiskin-takaa-ratkaisi#.VBwhYp09F2k.facebook ) the article of Zhang's ...
3
votes
2answers
62 views

partitions and their generating functions and Partitions of n

A partition of an integer, n, is one way of writing n as the sum of positive integers where the order of the addends (terms being added) does not matter. p(n, k) = number of partitions of n with k ...
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0answers
52 views

[ANSWERED]Is $\{n, n^{2} n^{3}\}$ a group under multiplication modulo $m = n + n^{2} + n^{3}$?

My number theory has been lacking, so i decided to practice it a bit. I have gotten better in the sense that i can figure out where to begin approaching a problem, but i am having trouble seeing the ...
3
votes
1answer
163 views

units in discrete valuation rings

Let $B/A$ be an extension of discrete valuation rings such that the purely inseparable extension of residue fields $l/k$ has a primitive element, i.e., $l=k(y)$ for some element $y$ in $l$. I want to ...
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votes
1answer
24 views

Gcd of two expressions

Given $$a(n) =n^2+20$$ Find the possible values of $$\gcd( a(n), a(n+1) ).$$ I tried doing this and got that the $\gcd$ of both the numbers should divide $2n+1$, but after this I am not able to get ...
2
votes
1answer
39 views

Concerning types of square-free numbers and comparing sizes of their subsets.

Call a square-free a 2-prime if it has exactly two prime divisors. Call a square-free a 3-prime if it has exactly three prime divisors,etc. Does there exist an integer sufficiently large N such that ...
0
votes
1answer
28 views

Hilberts Theorem (norm group)

The theorem says the following: The map $N$ is a group homomorphisim from the multiplicative group of $\mathbb{Q}^{x}[i]$ to the multiplicative group of $\mathbb{Q}^{x}$ and has kernel $\lbrace ...
-4
votes
1answer
33 views

If $a$ and $b$ are positive integers, then $a^{1/b}$ is either irrational or an integer. [on hold]

Please prove: If $a$ and $b$ are positive integers, then $a^{1/b}$ is either irrational or an integer.Thank you.
0
votes
1answer
39 views

Integer $k$ such that $k!$ has 99 zeros

For how many positive integers $k$ does $k!$ has 99 zeros. The question is not difficult,since if $k$ the first for $k!$ to have 99 zeros, then since $k+1,\cdots,k+4$ are not divided by 5, so the ...
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2answers
55 views

Making 24 with given number N

Initially we have a sequence of n integers: 1, 2, ..., n. In a single step, we can pick two of them, let's denote them a and b, erase them from the sequence, and append to the sequence either a + b, ...
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1answer
45 views

Approximation of a rational number

I have been asked the following problem: Every real number $x$ can be written as a sum of the form $$ \sum_{i=1}^{\infty} \frac{a_n}{n^2}, $$ where $a_1\in \mathbb{Z}$, $a_2=0,1,2,3$, and $0\leq ...
0
votes
2answers
23 views

powers of coprime numbers

if p and q are coprime integers does that mean the positive integral powers of p and q are coprime as well? E.G if p and q are coprime integers does that imply p^3 and q^3 are also coprime?
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0answers
55 views

Mordell Diophantine: $x^2+11=y^3$

I've been trying to solve the diophantine $$x^2+11=y^3$$ recently but to no avail. I tried the "UFD trick", re-writing as $(x-i\sqrt{11})(x+i\sqrt{11})=y^3$, but it didn't give me all the solutions. I ...
0
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1answer
33 views

expository articles on special values of L functions

While searching for some notes on L functions i have seen the following statement... In mathematics, the study of special values of L-functions is a sub field of number theory devoted to generalizing ...
2
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0answers
19 views

How prove that there are $a,b,c$ such that $a \in A, b \in B, c \in C$ and $a,b,c$ (with approriate order) is a arithmetic sequence?

Let $N=\{ 1,2,3,..., 3n \}$ with $n$ is a positive integer and $A,B,C$ are three arbitrary sets such that $A \cup B \cup C = N, A \cap B = B \cap C = C \cap A = \varnothing, |A| = |B| = |C| = n $. How ...
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0answers
34 views

Finding a point on an elliptic curve

I have an elliptic curve with the equation $ y^2 = x^3 + ax + b $ in modulo p, where p is prime. I also have a point G on that curve. How can I find another point that isn't a multiple of G? I ...
1
vote
3answers
92 views

Sum of the digits

Let $N$ be the greatest number that will divide $1305,4665$ and $6905$, leaving the same remainder in each case. Then what is the sum of the digits in $N$?
1
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0answers
30 views

Punctured Elliptic Curve

I've come across the word "punctured elliptic curve" here and there, but none of the basic texts on the topic (Husemoller, Silverman) define or mention it. What point is removed from the curve (the ...
0
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1answer
23 views

Proving an inequality on the sum of $\log$ of primes.

Let $S(x)=\sum_{p\leq x} \ln(p)$ where $\sum_{p\leq x}$ denotes a summation over the positive prime numbers that are $\leq x$ Prove that $\forall n \in \mathbb N, S(2n+1)-S(n+1)\leq ...
3
votes
2answers
44 views

Can you derive a formula for the semiprime counting function from the prime number theorem?

E.g., there are $4$ semiprimes less than or equal to $10$ $(4, 6, 9, 10)$ or $2$ squarefree semiprimes ($6$ and $10$). It's ok if it's off for small numbers but gets more accurate as $n \to \infty$.
2
votes
0answers
31 views

Partition function proof

I am looking for any online information regarding Hardy and Ramanujan's proof, perhaps the proof itself, that the partition function $p(n)$ is asymptotic to $$\frac{e^{K\sqrt{n}}}{4n\sqrt{3}}$$ where ...
0
votes
1answer
28 views

How can we show that $\pi (x) \leq \frac{x}{2}+1$?

What is the proof that the prime counting function $\pi (x)$ is such that $$\pi (x) \leq \frac{x}{2}+1$$
6
votes
4answers
854 views

Can any two irrational numbers NOT of the form (m+A) and (n-A) be added to produce a rational number?

$m$ and $n$ being rational numbers, A being an irrational number. I was wondering if two irrational numbers when added always yield an irrational number. All the counter-examples I could find were of ...
5
votes
3answers
239 views

What is the smallest natural number n?

What is the smallest natural number n for which there is a natural k, such that, the lasts 2012 digit in the representation decimal of $n^k$ are equal to 1? I don't even know how to start with it ... ...
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votes
0answers
63 views

Prove that $\sqrt{n}$ is irrational [on hold]

Question: Using fundamental theorem of integers and the fact that every natural number that is not prime, prove that $\sqrt{n}$ is irrational unless $n=m^2$ for some $m\in\mathbb N$. Here is how I ...
0
votes
1answer
63 views

Fermat's Last Equation

Sorry this is an amateur question but I was wondering since Andrew Wiles solved Fermat's Last Theorem what effect does this have any impact on Geometry. Does this prove in a sense Higher Order right ...
17
votes
1answer
413 views

Find $a,n\in \mathbb N^{+}:a!+\dfrac{n!}{a!}=x^2,x\in \mathbb N$

Find $a,n\in \mathbb N^{+}:a!+\dfrac{n!}{a!}=x^2,x\in \mathbb N.$ I find $\{n,a\}=\{4,1\}\{4,4\}\{5,1\}\{5,5\}\{7,1\}\{7,7\}\{20,11\}.$ (These are all if $n<300$.) ...
9
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0answers
69 views

Using properties of $\pi$ in the primes

We all know that $\pi$ and the prime numbers are intricately related, thanks to the work of famous mathematicians like Euler and Riemann. The irrationality of $\pi$ can ultimately be used to prove the ...
0
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0answers
30 views

Upper bound number of distinct prime factors

I want to prove that if $\omega (n)$ is the number of distinct prime factors of $n$ for $n>2$ we have $\omega (n) \leq \frac{\ln n}{\ln \ln n} + O(\frac{\ln n}{(\ln \ln n)^2})$. How can I do this? ...
0
votes
0answers
10 views

Name/properties of a difference of continuants

Suppose that $q_1$, $\ldots$, $q_s$ is a sequence of positive integers. Denote by $[q_1, \ldots, q_s]$ the numerator (in lowest terms) of the rational number represented by the continued fraction $$ ...