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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Number Theory Problem involving fractional part of a number

If $x = ( 9 + 4 \sqrt {5} )^{48}$ where $x = [x] + f$, where $[x]$ is he integral part of $x$ , and $x$ is its fractional part How do I go about finding the value of $x(1-f)$ ? Thanks!
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1answer
52 views

How come $\ n\ $ always divides at least one of the item of the sequence?

Given positive integer$\ \displaystyle n,\ $ the sequence is: $\displaystyle 2^n$ $\displaystyle 2^n - 2^{n-1}$ $\displaystyle 2^n - 2^{n-1} + 2^{n-2}$ $\displaystyle 2^n - 2^{n-1} + 2^{n-2} - ...
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2answers
26 views

To calculate the remainder of (111…) + (222…) + (333…) + (444…) + (555…) + (666…) +(777…) by 37

To Evaluate the remainder Question: $ (111...) + (222...) + (333...) + (444...) + (555...) + (666...) +(777...)$ mod $37$ In each bracket, the single digit $(1, 2, 3, ..., 7)$ is written $110$ ...
0
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1answer
18 views

something similar to the Bézout's identity, but with three integers.

There are three positive integers,not all equal. And their greatest common divisor is 1. We can perform this operation on them: choose two not equal integer $a,b(a<b)$ from them, and then ...
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3answers
92 views

Why are $e$ and $\pi$ believed to be normal?

I've found that affirmation in several sources, but I can't think of an obvious reason.
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0answers
13 views

A modified Shapiro's Tauberian theorem? Proof or counterexample

Let $\{a(n)\}$ be a nonnegative sequence such that $$\sum_{n\leq x}a(n)[x/n^{2}]=x^{2}\log x + O(x^{2})$$ for all $x\geq 1$, where $[y]$ denotes the greatest integer $\leq y$. Is true that the ...
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2answers
43 views

fermat's little theorem prove [on hold]

Prove that the third number of fermat's is prime? any help with the prove ? I meant prove that $257$ is prime
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2answers
41 views

Circular table problem

I've looked other questions that might help solve my problem, but haven't found any people who've used my method to solve it. The problem goes like this: Suppose there are 7 men and 5 women, and they ...
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0answers
23 views

Modified arithmetical functions from a modified Möbius function

In this post when $n>1$ we assume that $n=\prod_{k=1}^{\omega(n)}p_{n}^{e_{k}}$ its factorization in prime powers, where $\omega (n)$ is the number of distinct primes. It is well known the ...
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0answers
26 views

proof of chinese remainder theorem $x=a_1M_1y_1+…+a_nM_ny_n$?

I can't understand the proof of Chinese Remainder Theorem let $x ≡ a_1 (\text{mod }m_1 ),$ $x ≡ a_2 (\text{mod }m_2 ),$ · · · $x ≡ a_n (\text{mod }m_n )$ such that $m_1,m_2,...,m_n$ are relatively ...
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0answers
63 views

What do we know about the first occurrences of prime gaps?

Are there any conjectures from which we can infer something about the first occurrences of prime gaps length $n$ and their distribution? I've made an interesting graph of these values to make this ...
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0answers
39 views

A number theory problem. [on hold]

If $\gcd(a,b) =1$, prove that $\gcd(a-b+bm, a-b+bn) = 1$ where $n= a + bm$.
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1answer
40 views

Why is the Bernoulli Number $B_1$ sometimes $+ \frac{1}{2} $?

By using the recursive formula, \begin{equation} \sum_{i=0}^{n} \binom{n+1}{i} B_i = n+1 \end{equation} we find $B_1$ to be $\frac{1}{2}$ and not $- \frac{1}{2}$. Why is this?
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1answer
171 views

a new continued fraction for $\sqrt{2}$

In a q-continued fraction related to the octahedral group I defined a new q-continued fraction of the squared of ramanujan's octic continued fraction which I discovered using certain three term ...
4
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0answers
29 views

Dividing the whole into a minimal amount of parts to equally distribute it between different groups.

Suppose we have a finite amount of numbers $x_1, x_2, ..., x_n$ ($x_i\in\mathbb{N}$) and an object that should be divided into parts in such a way that it can be without further dividing distributed ...
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1answer
26 views

question about first occurring prime gaps

If a prime gap $g(p)$ is the first occurring prime gap of it's size, does this imply that it is also the largest gap below $p$? In other words, is the set of first occurring prime gaps contained ...
4
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1answer
32 views

Pair of Circles Intersect

If $S$ is a collection of circles s.t. for each point $c$ on the x-axis there is a circle in $S$ passing through the point $(c,0)$ and at the same time has the x-axis as a tangent to the circle at ...
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0answers
53 views

Algorithm to answer existential questions - Reduction

Lemma 1. For any $x$ in the ring $F[t,t^{-1}]$ ($F[t,t^{-1}]$: the polynomials in $t$ and $t^{-1}$ with coefficients in the field $F$), $x$ is a power of $t$ if and only if $x$ divides $1$ and ...
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0answers
75 views

Are there papers or books that explain why Bernhard Riemann believed that his hypothesis is true?

I would like to know what are the mathematical reasons for which Bernhard Riemann believed that his hypothesis is true, and I would like to know if those mathematical reasons were cited in his ...
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2answers
50 views

Permutations minus Transpositions

I want a formula that allows me to find all the permutations in $S_n$ (which is the set of all the integers from 1 to $n$) which don't contain a transposition. Attempt: Lets call $g(n)$ the ...
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1answer
66 views

algebra question.. [on hold]

If $f : \mathbb{R}\rightarrow \mathbb{R}$, and $f(x)=\frac{2}{4^{x}+2}$ Find the value of $$f\left [ \frac{1}{11} \right ]+f\left [ \frac{2}{11} \right ]+ \cdots +f\left [ \frac{10}{11} \right ]$$
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2answers
50 views

Terms of a certain recurrence

Let $a_1, a_2\dots $ be a sequence of reals such that $a_1 = a_2 = 1$, and $$a_{n + 2} = \frac{a_{n + 1}^3 + 1}{a_n}$$ for $n \ge 1$. It appears to be the case that all of these values are integers. ...
3
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1answer
68 views

Frobenius automorphism and non-split Cartan subgroup of $\mathrm{GL}_2(\mathbb{F}_p)$

For completeness I give some definitions. Let $p$ a prime number and consider $V$ a 2-dimensional $\mathbb{F}_p$-vector space. Consider $k$ a sub-algebra of $\mathrm{End}(V)$ that is a field of $p^2$ ...
3
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2answers
60 views

Indexes of prime Fibonacci numbers

I found this on Mathworld, but I can't seem to find any proof, either on StackExchange, nor any other site: Why do all Fibonacci primes, except for $F_4=3$, have prime indexes (with $F_0=0$)? My ...
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1answer
28 views

Fermat primality test and Fermat pseudoprime

What is the difference between Fermat primality test and Fermat pseudoprime?Can anyone explain me how we use them ?
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0answers
62 views

Zero-Sum Partitions of Nonzero Elements of a Ring

In this question, rings are not necessarily finite nor do they need to be unital (i.e., the multiplicative identity may not exist). Although I shall exclusively discuss finite commutative unital ...
0
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1answer
23 views

Combining Moebius transformations

Moebius transformation in this case $\frac{az+b}{cz+d}$ for complex $z$. I have several transformations I want to apply to an initial $z$. For example first transform $f(a,b,z) = z + (a + bi) = ...
3
votes
1answer
44 views

FRACTRAN for natural numbers

Is there a simple analogue of FRACTRAN that maps a natural number to a natural number, instead of mapping a list of fractions to a natural number? One could use Gödel encoding to translate FRACTRAN ...
0
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3answers
27 views

Find the number of seven digit whole numbers in which only 2 and 3 are present as digits if no two 2's are consecutive in any number?

Find the number of seven digit whole numbers in which only $2$ and $3$ are present as digits if no two $2$'s are consecutive in any number? My Approach: We can make numbers and see like: ...
0
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1answer
31 views

If $500! = 2^m\cdot$N, where N is an odd positive integer, then find $m$

Problem : If $500! = 2^m\cdot$N, where N is an odd positive integer, then find $m$ My approach : Shall we need to expand $500!$ and then find prime factors and see what is the power of 2 in that ...
10
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1answer
122 views

why is $\zeta(1+it) \neq 0$ equivalent to the prime number theorem?

Reading through Titchmarh's book on the Riemann Zeta Function, chapter 3 discusses the Prime Number Theorem. One way to prove this result is to check the zeta function has no zeros on the line $z = 1 ...
3
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1answer
16 views

Non-additive asymptotic upper density: $\mathsf{d}^\star(A\cup B) \neq \mathsf{d}^\star(A)+\mathsf{d}^\star(B)$

Let $\mathsf{d}^\star$ be the asymptotic upper density on $\mathbf{N}$, that is, for each $X\subseteq \mathbf{N}$ we have $\mathsf{d}^\star(X)=\limsup_n |X\cap [1,n]|/n$. Then, is it possible to ...
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0answers
65 views

Sets with $n$ prime numbers

Let $n>2$ a natural number. We define the following sets: $$S=\{1 \leq a \leq n : (a,n)=1, a^{n-1} \not\equiv 1\pmod n\} \\ T=\{1 \leq b \leq n : (b,n)=1, b^{n-1} \equiv 1 \pmod n\}$$ ...
1
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1answer
31 views

A question on arithmetic progressions

Is it true that for every $n \in \mathbb N$ , $\exists N \in \mathbb N$ such that for any subset $A \subseteq \{1,2,...,N\}$ , either $A$ or $\{1,2,..,N\} \setminus A$ contains an arithmetic ...
11
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1answer
82 views

Number of polynomial factors of $a^n-b^n$?

This is a number theoretical problem that I discovered myself. Let $f(n)$ be the number of factors of $a^n-b^n$ with integer coefficients when its completely factored. For example: $f(1)=1$, because ...
8
votes
4answers
129 views

Show that $t^n-1 \mid t^m-1 \Leftrightarrow n\mid m$

I want to prove the following lemma: $t^n-1$ divides $t^m-1$ in $F[t, t^{-1}]$ if and only if $n$ divides $m$ in $\mathbb{Z}$. I have done the following: $\Leftarrow $ : $n\mid m \Rightarrow ...
0
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0answers
29 views

How can we show the assertion?

Every natural number not of the form $4^n(8m+7)$ where $m$ and $n$ are natural numbers, can be represented as sum of three squares.
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1answer
37 views

Good introductory book for Probabilistic Number Theory

I have a decent high school knowledge of Elementary Number Theory and it is also a subject I love to study. I have a good background in Real Analysis (not Complex Analysis) and Abstract Algbera. I ...
3
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0answers
71 views

Solutions to $\prod_{i=1}^{k}p_i=\sum_{i=1}^{k}p_i^2$ with $p_i$ distincts primes

Does $\prod_{i=1}^{k}p_i=\sum_{i=1}^{k}p_i^2$ with $p_i$ distincts primes and $k\geq2$ have a solution ? Here is what I already know : There is no solutions if $k\equiv0\bmod2$ or if ...
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1answer
69 views

Other solutions to 1+1 = 2 [on hold]

Under the usual assumptions of maths and the definition of numbers we have 1 + 1 = 2. However if someone is imaginative enough we can find that: 1 + 1 = 1 (Boolean Algebra) 1 + 1 = 10 (Binary ...
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0answers
23 views

Calculating $n$-th $q:P(q)=p \in \Bbb P$

Let $P(x)$ denote the number of ways of writing an integer $x$ as a sum of positive integers (where permutation of the array of integers in the sum doesn't count). Ex: $P(1)=1, P(2)=2,P(4)=5$. Let ...
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1answer
31 views

Harmonic Identity Evidence

I was recently cleaning up my laptop when I stumbled upon a spreadsheet that I created a while ago at school when I still had an interest in math. Anyway, I remember reaching the following identity ...
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1answer
29 views

Quadratic reciprocity in the case $a=-1$

I am reading the proof the for odd prime $p$, $$ \left ( \frac{-1}{p} \right)_2 = (-1)^{\frac{p-1}{2}} = \begin{cases} 1 \hspace{2mm} \text{for} \hspace{2mm} p \equiv 1 \operatorname{mod} 4 \\ -1 ...
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0answers
38 views
+50

Let $f(x)$ be defined over all rationals $x$ in $[0,1]$ and let $F(n) = \sum_{i=1}^n f(\frac in)$

also define $$F^*(n) = \sum_{i=1\,\,(i,n)=1}^n f(\frac in)$$ then prove that $$F^* = \mu * F$$ where $\mu$ is the Möebius function and the $*$ means the Dirichlet convolution. I tried the Bell series ...
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1answer
54 views

Block of integers: Divisibility

Let a < b be natural numbers. Prove that every block of b consecutive natural numbers contains two distinct elements whose product is divisible by ab. (I've proved this) Suppose now a < b < ...
3
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1answer
62 views

Find the sum of all primes smaller than a big number

I need to write a program that calculates the sum of all primes smaller than a given number $N$ ($10^{10} \leq N \leq 10^{14} $). Obviously, the program should run in a reasonable time, so $O(N)$ is ...
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0answers
34 views

If $p\mid 2^n\pm1$ with $p$ and $n$ relatively prime, then $p$ is a Wieferich prime iff $p^2$ also divides $2^n\pm 1$

The Wolfram Mathworld article on Wieferich primes states: $2^{p-1}-1\equiv 0 \mod p.$ If the first case of Fermat's last theorem is false for exponent $p$, then $p$ must be a Wieferich prime ...
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1answer
40 views

Chinese Remainder Theorem example

$$x = 4 \bmod 18$$ $$x = 52 \bmod 96$$ $$x = 6 \bmod 20$$ My current algorithm thinks the answer is $x \equiv 1066 \bmod 1440$ but I don't think there should be a solution to this. The algorithm: ...
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0answers
39 views

Proofs needed for observations regarding prime-partitionable numbers.

Definition of a prime-partitionable integer taken from W. Holsztynski, R. F. E. Strube, Paths and circuits in finite groups, Discr. Math. 22 (1987) 263-272, doi and apparently the same as in W. T. ...
2
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1answer
52 views

solutions for Diophantine equation for ${k_1} + 2{k_2} + 3{k_3} + … + n{k_n} = n$

Consider the Diophantine equation of the form ${k_1} + 2{k_2} + 3{k_3} + .... + n{k_n} = n$, where ${k_1},{k_2},...{k_n} \in Z^+$ . For a given $n$, how can I obtain the solutions of a given equation? ...