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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4
votes
2answers
102 views

Minimum number of guesses on sum and product required to find two numbers.

I have a series of numbers 1 to N. A system randomly picks up two numbers and computes their sum and product. I have to guess the sum and product, The system will tell if the sum and product are ...
2
votes
2answers
251 views

Firoozbakht's conjecture and maximal gaps

In the Wikipedia article, it seems to me as if it's implied that it is enough to check the conjecture only for maximal gaps (numbers $n$ s.t. $\forall k<n:g_n>g_k$). I.e it holds that ...
0
votes
2answers
48 views

Choosing sets yielding large sumsets

I am working on a problem which basically boils down to wanting to choose sets $A_1, \ldots, A_m$ over a group $G=(\mathbb{Z}_C,+)$, where $C$ and $m$ are constants, such that the sumsets defined with ...
0
votes
1answer
66 views

$\left| (4 \mathbb{N} -1) \cap \mathbb{P} \right| \ = \ \infty$ where $\mathbb{P}$ is the set of prime numbers. [duplicate]

I try to show that there are infinitely many prime numbers in the set $ \{ 4n-1 \ : \ n \in \mathbb{N} \}$. I've been told that I needed to adjust Euclid's proof a bit so that it would work for ...
0
votes
2answers
69 views

Euler's function Phi

Which of the following statement is/are true? $\phi$$(n)$ is even as many times as it is odd. $\phi$$(n)$ is odd for only two values of $n$. $\phi$$(n)$ is even when $n>2$. $\phi$$(n)$ is odd ...
2
votes
2answers
246 views

Solutions of Diophantine equation

Does there exists any other solutions of the following Diophantine equation $$zx^2 +xy^2 +yz^2 =xyzt .$$ I found that $$(x,y,z,t) =(s,s,s,3) ,(x,y,z,t)=(s,2s,4s ,5)$$ where $s\in\mathbb{N}$ are ...
3
votes
0answers
66 views

Bessel function sum?

Is $$f(x)=\dfrac{2\pi}{n^2}\sum_{m=1}^{x}\sum_{k=1}^{n}\dfrac{\Im(e^{\dfrac{\pi i m}{\Im(\rho_k)}})k}{\Im(\rho_k)}$$ related to the Bessel family of functions? Plot for $n=300$ Or is it related ...
1
vote
1answer
42 views

Multiplicative submonoid: an exercise

Here's my problem: (a) Show that the set of integers, which can be written as $a^2+ab+b^2$ for some $a,b \in \mathbb{Z}$ is a multiplicative submonoid of $\mathbb{Z}$; (b) Explain how all ...
1
vote
1answer
66 views

Help understanding proof of Legendre's formula

Can somebody kindly help me in understanding below highlighted line in proof of Legendre's formula Particularly this step : $$ \sum\limits_{i=1}^{\infty}\left\lfloor\dfrac{n}{p^i}\right\rfloor = ...
2
votes
0answers
75 views

Numbers which are writable as a sum of permutation pairs

We say that $N$ is writable as a sum of permutation pair $\{a,b\}$ if $a+b=N$, $a\neq b$ and $a$ and $b$ are permutations of each other (e.g. $321 = 156 + 165 = 147 + 174 = ... $). Looking at 3-digit ...
1
vote
0answers
50 views

How prove this sum indentity with the zeta function and divisor functions

show that $$\zeta{(s)}\zeta{(s-a)}\zeta{(s-b)}\zeta{(s-c)}\zeta{(s-a-b)}\zeta{(s-b-c)}\zeta{(s-a-c)}\zeta{(s-a-b-c)} ...
1
vote
1answer
74 views

On some inequalities which are an generalization of F. Beukers' corresponding results

In 1981, F. Beukers proved the following theorem in his article On the generalized Ramanujan-Nagell equation I : Theorem $\bf1$. $~$ Suppose $m \in \mathbb{Z}$, then for any integer $x$, $$ ...
0
votes
2answers
267 views

How find all positive real $\beta$ such A finite number of $\left|\frac{p}{q}-\sqrt{2}\right|<\frac{\beta}{q^2}$

Define sequence $\{a_{n}\}$,such $$a_{1}=1,a_{2}=2,a_{k+2}=2a_{k+1}+a_{k},k\ge 1$$ Find all positive real number $\beta$,such only have a finite number of relatively prime integers $(p,q)$ ...
2
votes
2answers
172 views

How find the value $\beta$ such $\left|\frac{p}{q}-\sqrt{2}\right|<\frac{\beta}{q^2}$

Find all positive real number $\beta$,there are infinitely many relatively prime integers $(p,q)$ such that $$\left|\dfrac{p}{q}-\sqrt{2}\right|<\dfrac{\beta}{q^2}$$ maybe this problem background ...
1
vote
0answers
88 views

show that $\sum\limits_{d|n} \lambda(n/d)2^{\omega(d)}=1$

show that $\sum\limits_{d|n} \lambda(n/d)2^{\omega(d)}=1$ $\lambda$ is Liouville_function $\omega$ is number of distinct prime factors function I have proved $\lambda$ to be multiplicative in ...
0
votes
1answer
90 views

Using Euler's Theorem to find remainders

I am asked to find $34^{82248}$ mod $(83)$.I get this down to $34^{5}$ mod $(83),$ but stuck here. I am assuming acquiring the answer should require computing something like $34^{5}$. Thanks in ...
2
votes
2answers
175 views

Legendre's formula

Legendre's formula counts the number of positive integers less than or equal to a number $x$ which are not divisible by any of the first $a$ primes: $$\begin{align} &\phi(x,a)=\lfloor x ...
0
votes
2answers
35 views

Can index in a radix be $1$?

I am stuck at a very basic question I had a true/false question where the statement was "if $n$ is a natural number and $x$ a prime number $$\sqrt[n]{x}$$ is always irrational " I was of the ...
1
vote
1answer
89 views

Is there a short proof of the formula for Legendre symbol $(\frac{2}{p})=(-1)^{(p^2-1)/8}$?

Let $p$ > 2 be a prime number. I found in wiki a complex proof for this Legendre symbol: $$\left(\frac{2}{p}\right) = (-1)^{\frac{(p^{2}-1)}{8}}$$ Can anyone give me a short solution please?
3
votes
2answers
162 views

Find the 6 Digit Number

$N$ is a 6 digit Natural number such that its sum of the digits is $43$. Find $N$ if Exactly One of the statements below is False: $(1)$ $N$ is a Perfect Square $(2)$ $N$ is a Perfect Cube $(3)$ N ...
3
votes
1answer
325 views

Shtukas?$\mbox{}$

Does there exist an exposition of the significance of shtukas for someone who is mathematically literate but is largelly ignorant of Drinfeld modules? This arises in the work of Peter Scholze among ...
2
votes
0answers
63 views

Bounds re Asymptotic Formula for the Sum of Largest Prime Factors

I have a reference request related to the result : $\sum_{n=2}^{x} P(n)$ ~ $\frac{\pi^2}{12}\frac{x^{2}}{log(x)}$ as $x \rightarrow \infty$ where $P(n)$ is the largest prime factor of the positive ...
0
votes
4answers
79 views

Solve $c^2-b^2-a^2=2N$

Is there anyone that can help solving this equation: $c^2-b^2-a^2=2N$ where $a,b,c,N$ are natural numbers. Edit: We need to express $a,b,c$ for a certain $N$. Regards
4
votes
2answers
211 views

what is the most easy to read Algebraic Geometry book? [duplicate]

All: what is the most easy to read (most accessible) Algebraic Geometry book ? (If possible, I am looking for an introduction book, maybe for undergraduate, and maybe similar to A Friendly ...
1
vote
4answers
134 views

find the largest four digit prime number ending with $53$

I have started something like below but not able to make any progress : Say the number is $ab53$ $1000a+100b + 53 = p$ $1000a+100b = p-53$ that means $100 | p-53$
4
votes
1answer
108 views

Asymptotics on the largest prime for which $x^n+1\equiv y^n$ has no nonzero solution

It $\let\epsilon\varepsilon\let\leq\leqslant\let\geq\geqslant$is a well known result that for every $n\in\mathbb N$, $x^n+1\equiv y^n\pmod p$ is non-trivially solvable for sufficiently large primes ...
2
votes
1answer
121 views

Do all automorphisms of $\mathbb{H}$ preserve the norm of an element?

Do all automorphisms of $\mathbb{H}$-- the Hamilton quaternions-- preserve the norm of an element? I can't seem to answer this question without using the not-so-elementary fact that all automorphisms ...
0
votes
1answer
43 views

odd squarefree and squareful neighbors

There are squarefree numbers $n=\prod_{i=1}^{k}p_i$ so that $n+2$ is not squarefree (e.g. $115+2=3^2.13$). Are there infinite many such $n$? Are there numbers n with arbitrarely many prime-factors? ...
1
vote
0answers
89 views

How many zero's does a general real entire function $f(z)$ have?

Let $f(z)$ be a real entire function. How do we find the number of solutions for $f(w)=0$ ? Can we express the number of zero's of $f$ in terms of its Taylor coëfficiënts ? Im not looking for the ...
9
votes
1answer
3k views

Relationship between Primes and Fibonacci Sequence

I recently stumbled across an unexpected relationship between the prime numbers and the Fibonacci sequence. We know a lot about Fibonacci numbers but relatively little about primes, so this connection ...
0
votes
2answers
50 views

Proving the following relation

If $p$ and $q$ are distinct primes show that $$p^{q-1}+q^{p-1}=1 \bmod pq $$ In this question what I did was I wrote out the Fermat's relation and got $$p^{q-1}=1 \bmod q$$ $$q^{p-1}=1 \bmod p $$ ...
4
votes
6answers
136 views

Finite sequences of prime numbers

There is a lot of prime sequences: prime numbers in a special form. For example Mersenne primes are primes of the the form $2^n-1$, or Pythagorean prime are primes of the form $4n+1$. Even primes are ...
11
votes
7answers
860 views

Define an infinite subset of primes such that the sum of reciprocals converges

How can we define an infinite subset of primes such that the sum of reciprocals converges? $S=\{p\in \mathbb{Z}^+ : p\ \text{is prime and some condition on}\ p\}$ s.t. ...
6
votes
3answers
173 views

Does the sum of the reciprocals of all primes of the form $4k+1$ converge?

Let $S=\{p\in \mathbb{Z}^+ : p\ \text{is prime and}\ p\equiv 1 \mod \ 4\}.$ Is $\displaystyle\sum_{p\in S}\frac{1}{p}$ finite or infinite, and where can I find more information about it?
2
votes
1answer
360 views

Count numbers in a range “A to B” which the number of its divisors equal to N

I'am looking for efficient algorithm to find the number of divisors for Numbers in a Hugh Rang up to 10^9. Such task is presented in those two problems: NDIV, and spoj NFACTOR I used prime ...
2
votes
1answer
178 views

Modular forms are arithmetic objects

What does arithmetic object exactly means? In an article, I found the following statement: modular forms are arithmetic objects. What this should means? Bests.
0
votes
1answer
49 views

Yet another diophantine approximation question

Let $x=(x_1,\ldots,x_n)$ be a real vector in general position and let's say it is normalized: $\lVert x\rVert=1$. Let $y$ be a real number that can be arbitrarily large and $\epsilon>0$ can be ...
5
votes
3answers
255 views

Divisors of factorials

Let $p$ be a prime number and $k$ a positive integer. Let $d$ be the smallest positive integer such that $p^k$ divides $d!$. It is true that $d$ is necessarily a multiple of $p$?
6
votes
1answer
131 views

How prove $ y^2=x^3+x+1370^{1370}$ has at least 6 answers in $ \mathbb{Q}$?

How prove that $ y^2=x^3+x+1370^{1370}$ has at least 6 answers in $ \mathbb{Q}$?
1
vote
1answer
1k views

How to determine the key-matrix of a Hill cipher where the encrypted-message-matrix is not invertible?

I am new to this subject and I have a homework problem based on Hill cipher, where encryption is done on di-graphs (a pair of alphabets and not on individuals). The alphabet domain is $\{A\dots ...
2
votes
2answers
888 views

Chinese Remainder theorem with non-pairwise coprime moduli proof

There exists a $x \in \mathbb{Z}$ satisfying system of equations: $$x=a_1 \pmod {n_1}$$ $$x=a_2 \pmod {n_2}$$ $$\ldots$$ $$x=a_k \pmod{n_k}$$ if and only if $a_i=a_j \pmod{\gcd(n_i,n_j)}$ for all ...
1
vote
2answers
74 views

series for $n$-th prime number and prime counting function

"Theoretical Computer Science Cheat Sheet" gives the following: $$p_n = n \ln n + n \ln \ln n - n + n \frac{\ln \ln n}{\ln n} + \mathcal{O}\left( \frac{n}{\ln n}\right)$$ $$\pi (n) = \frac{n}{\ln n} + ...
2
votes
0answers
191 views

Are there infinite many primes p such that 2p-1 is also prime?

I did a search online and found a similar notion called Sophie Germain prime, which by definition is a prime $p$ such that $2p+1$ is also prime. Sophie Germain primes are conjectured to be of infinite ...
12
votes
4answers
5k views

Why are all non-prime numbers divisible by a prime number?

In Euclid's infinite prime numbers proof, the logic is as follows: Assume a set $S$ of all prime numbers in existence is finite (there are a finite amount of primes) Then there must be a greatest ...
4
votes
3answers
109 views

Prove that there are infinitely many primes $P_i\equiv1\pmod6$

Proving that there are infinitely many primes is fairly simple: Assume that there is a finite number of primes. Let $G$ be the set of all primes $P_1,P_2,\ldots,P_n$. Compute $K = P_1 \times P_2 ...
3
votes
2answers
121 views

Finding $23! 7! \bmod 29$ using Wilson's Theorem

I'm trying to reduce $23!\,7! \bmod 29$. I used Wilson's Theorem to get $23!(120)\equiv 1 \pmod{29}$. I then solved $120a\equiv 1 \pmod{29}$ and got $a\equiv 22$. I then computed $7! \pmod {29}$. ...
2
votes
3answers
120 views

Is there a name for this pattern?

I'm not a mathematician, but I was calculating multiplication of some numbers and I saw a pattern emerging. What is this phenomenon called? And does it happen in other cases? ...
5
votes
0answers
175 views

Diophantine: $x^3+5=y^5$

Find all integers $x$ and $y$ such that $x^3+5=y^5$. I found this after solving the equation $3^a+5=2^b$. For this equation, since $(a,b)=(3,5)$ is a solution, it is possible to write it as ...
3
votes
4answers
115 views

If $q^n$ is irrational for all $n>1$, then $q$ is irrational.

Theorem. Let $q \in \mathbb{R}$ an arbitrary given number. If $q^n$ is irrational for all $n>1$ integer, then $q$ is irrational. My Questions. What is a the name of this statement and what is the ...
10
votes
0answers
256 views

Has category theory solved major math problems?

All: I am new to category theory. Just wonder if category theory has solved any major math problems for other mathematics fields? or what are the major applications of the category theory ? ...