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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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
42 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
20 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
33 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 ...
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1answer
20 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) = ...
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0answers
25 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 ...
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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: ...
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1answer
28 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 ...
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35 views

I dont know about cryptography,Help me with cryptography [on hold]

I have just started my cryptography course. But do not have the idea how to start.give me some name of good books. I also want to know the number theory approach on Cryptography. plz help. THANKS
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1answer
119 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 ...
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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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60 views

Sets with $n$ prime numbers

Let $n>2$ a natural number. We define the following sets: $$S=\{1 \leq a \leq n : (a,n)=1, a^{n-1} \not\equiv 1\pmod n\} \\ T=\{1 \leq b \leq n : (b,n)=1, b^{n-1} \equiv 1 \pmod n\}$$ ...
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1answer
28 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 ...
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1answer
80 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 ...
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1answer
62 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 ...
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0answers
27 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
31 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 ...
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0answers
68 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
65 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
29 views

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
48 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 < ...
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1answer
59 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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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
38 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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37 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. ...
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1answer
50 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? ...
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2answers
419 views

Prime Numbers and a Two-Player Game

In this question, $\mathbb{N}_0$ is the set of all nonnegative integers. The notation $\mathbb{N}$ is reserved for the set of all positive integers. Alex and Beth are playing the following game. ...
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26 views

Properties of a Semi-modulo! operation

Let $A$ be an integer with its representation in base $p$ ($p$ may be prime number but not necessarily) described as: $$A=(a_ma_{m-1}\ldots a_1a_0)_{p}$$ We know $A\equiv (a_n\ldots a_1a_0)_{p}\pmod ...
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50 views

How is named this property of zero parity? [on hold]

How is named then this property of zero parity? Numbers parity Any number, except zero, multiplied twice, is an even number. A pair has two elements. Demonstration: One multiplied ...
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2answers
246 views

Inequality involving a product over the primes

Is there someone who is able to prove the following statement? $$\prod_{m=1}^n \dfrac{p_m-1}{p_m} \leq \dfrac{1}{\ln(n)}$$ for all integers $n >1$ where $p_m$ is the $m$-th prime number.
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2answers
130 views

Is $a=\frac{1992!-1}{3449\times 8627}$ a prime number?

Is $a=\dfrac{1992!-1}{3449\times 8627}$ a prime number ? This is a natural follow-up to that recent MSE question We know that $a$ has $5702$ digits and no prime divisor $<10^6$.
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1answer
66 views

What is the inverse function of gcd? [on hold]

Let $a,x,c \in\mathbb{Z}$. If $\gcd(a,x)=c$ where $a, c$ are constants and $x$ is a variable, then what values can $x$ take and how to find those values ?
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1answer
33 views

Is there an isomorphism $\mathbb{C}_p \cong \mathbb{C}_q$ for primes $p \neq q$?

Let $p \neq q$ be distinct primes. Is there a field isomorphism $\mathbb{C}_p \cong \mathbb{C}_q$? Is there a field isomorphism $\mathbb{C}_p \cong \mathbb{C}$? If such an isomorphism exists, given ...
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0answers
42 views

A question on the inequality $\bigl(\pi(x+y)\bigr)^2<4\pi(x)\pi(y)$

From the answer of this post it seems highly probable that the following problem can be proved, Show that for all sufficiently large $\min(x,y)$ we will have, ...
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1answer
39 views

Maximal $n$ such the the additive partition with a given product is unique.

Given $n$, there are many tuples with $a + b + c = n,0 < a < b < c$. For large $n$, different tuples may give the same products. E.g. $2+8+9=19=3+4+12,2\times8\times9=144=3\times4\times12$. ...
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1answer
42 views

Congruence using extended GCD

$$\eqalign{ x &\equiv 5 \mod 15\cr x &\equiv 8 \mod 21\cr}$$ The extended Euclidean algorithm gives $x≡50 \bmod 105$. I understand now that if we combine the two it implies ...
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2answers
48 views

Why is this congruence true?

$$\eqalign{ x &\equiv 5 \mod 15\cr x &\equiv 8 \mod 21\cr}$$ The extended Euclidean algorithm gives $x≡50 \bmod 105$. How/why? I am trying to understand how this is true when ...
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2answers
52 views

$2$-adic sequence converging to $\sqrt{-7}$.

I am trying to construct a sequence in $\mathbb Q_2$ that is formed of rational numbers and converges to $\sqrt{-7}$, to prove that $(\mathbb Q, |\cdot|_2)$ is not complete. My lecturer stated that ...
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0answers
34 views

A residue question in integers

Given $N\in\Bbb N$, is it possible to find $9$ positive integers $A_j,N_i$ with $j\in\{1,2,3\}$, $i\in\{1,2,3,4,5,6\}$ such that following holds? $(1)$ $N\log N < A_j < cN\log N$ at every $j$ ...
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1answer
32 views

Prove that $\sum_{t \vert n} d^3(t) = (\sum_{t \vert n}d(t))^2$ for all $n \in \mathbb{N}$ [duplicate]

here $d(n)$ counts the number of positive divisors of $n$. I've tried 2 things: Using Bell series. But then again it just showed me that the bell series of the square of a function is not the ...
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0answers
22 views

Is there a solution to this problem on Fermat's Quotient?

We define Fermat's Quotient as $q_a = (a^p-1)/p \pmod p $ where $p$ is a prime greater than $2$. How will you prove that the only solutions of the equation $q_a=0$, $q_b=0$ and $q_{a+b}=0$ where ...
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0answers
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Need my CRT work spot-checked

So I have a bunch of equations that look like this: $$k + tx \equiv a \bmod m$$ Where $t$ is the common variable I am solving for among the equations (each equation may have different values for ...
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1answer
28 views

Why does this imply a congruence does not exist?

Consider $$kx \equiv a \bmod M$$ Where $x,a,M$ are known, solving for $k$. Let $g = \gcd(M,x)$. Why is it the case that if $g$ does not divide $a$, there is no solution?
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1answer
96 views

Possible proof of Fermat's Last Theorem for prime exponents greater than 2

I would appreciate if someone could check my attempt in proving the Fermat's Last Theorem for prime exponents greater than $2$. Firstly, let's prove a couple of lemmas which state that sum or ...
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0answers
36 views

Why is it so hard to find a generating function for Somos' sequence?

The sequence is $\{1,2,12,576,1658880,\dots\}$. The $n$th number is obtained by squaring the $(n-1)$-th number and multiplying by $n$. So we start with $a_1=1$, $a_2=1^22=2$, $a_3=(1^22)^23=12$. In ...
4
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0answers
99 views

rationality of $\ell$-adic representation attached to an elliptic curves

Let $E$ be an elliptic curves defined over a number field $K$. Consider the $\ell$-adic representation attached to $E$ $$ \rho_{\ell}:\mathrm{Gal}(\overline{K}/K) \longrightarrow ...
4
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1answer
61 views

Question about Mersenne numbers

We don't have any proof for Mersenne conjectures, but is it true that there exist infinitely many primes $p$ such that $2^p-1$ is not prime?