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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31 views

Is this a good generating function for Sum-of-divisors function?

I have an expression for the sum-of-divisors function defined as $$\sigma(n)=\sum_{d\mid n}d.$$ However I do not know how nontrivial or practical it actually is. Let us define ...
2
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0answers
7 views

Wireless networks on two sequential office floors: Random partitions of a finite interval via a point process on a line

Construct a Poisson point process of density one on a line of length $L$. Allow each point in the process to "see" part of the line to their left, and part of the line the their right (such that the ...
2
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0answers
43 views

“Race” of the primes modulo $1,3,7,9\ \pmod {10}$

The "race" starts with the prime $11$. The number of primes $1, 3, 7, 9 \pmod {10}$ is denoted after every occurring prime. Does the lead change infinitely often? And does every "runner" have ...
4
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0answers
35 views

$\tau$ and grouping of prime numbers

From Prime Number Theorem and this we can state $$\frac{p_n}{\bar{p}}\sim 2$$ or $$\lim_{n\to \infty} \frac{np_n}{(p_1 + \dots +p_n)} = 2$$ If we then look at the fluctuations in the graph of $$f(n) ...
2
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1answer
50 views

Prove that $\sum_{t=1}^{p-1} \frac{t^2-1}{t^2+1} \equiv 0 \pmod p$

I'm trying to prove the statement $\sum_{t=1}^{p-1} \frac{t^2-1}{t^2+1} \equiv 0 \mod p$ and I don't really know where to start. Obviously $\sum_{t=1}^{p-1} \frac{t^2-1}{t^2+1} = 2\sum_{t=1}^{(p-1)/2} ...
3
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1answer
27 views

Prove $v,w\in Z_{p}\times Z_{p}$ is linearly independent when $p=2$ and dependent when $p=3$

I need to prove that $\{v=(6,9),w=(7,8)\}\in Z_{p}\times Z_{p}$ is linearly independent when $p=2$ and linearly dependent when $p=3$. The problem is my freshman algebra course did not cover rings and ...
4
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2answers
141 views

a certain simple continued fraction

Given the golden ratio: $$\phi=\frac{1+\sqrt{5}}{2}$$ and the following simple continued fraction: $$G(q,k)=\cfrac{1}{1-q+\cfrac{1}{1-{q^3}^k+\cfrac{1}{1-{q^5}^k+\cfrac{1}{1-{q^7}^k+\ddots}}}}$$ For ...
3
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1answer
28 views

the least $m$ such that $a^m\equiv 1 \mod n $ for fixed $a,n$.

Is there any known method for calculating $\lambda_a(n)$ which returns the smallest integer $m$ such that $a^m\equiv 1 \pmod n$ where $\gcd(a,n)=1$ ? I searched but I found nothing, is there at ...
2
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1answer
30 views

Is there a finite initial generating set for ${\mathbb N}$ given these two operations?

This is inspired by a recent question. Suppose we have $x_j \in \mathbb{N}$, and then we are allowed to perform any sequence of the following two operations: 1) Multiply by $k$ for some fixed $k \in ...
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1answer
22 views

What should be the approach for finding the remainder

How can one approach this kind of question: Find the remainder when $\left((7!)^{6!}\right)^{17777}$ is divided by 17
6
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0answers
45 views

Riemann zeta function Euler product for primes equivalent to $3$ mod $4$

Question: can $$ \zeta_1(s) = \prod_{p \equiv 3 \pmod{4}} \frac{1}{1 - p^{-s}} $$ be evaluated or written in terms of standard functions? Details: We can write the Riemann zeta function as ...
1
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1answer
188 views

Would this proof strategy work for proving the lonely runner conjecture?

The problem is the lonely runner conjecture. This conjecture states that if $k$ runners begin running down a circle of unit circumference with random speeds, it will always the case that all runners ...
2
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2answers
53 views

Solving a quadratic Diophantine equation

I want to solve the following quadratic Diophantine equation: $$\frac{x(x-1)}{y(y-1)}=\frac{p}{q} \hspace{5 mm}, \hspace{5 mm}p\le q$$ For $p=1$ and $q=2$, it is easy to solve. Let $y=x+z$. Then ...
0
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1answer
17 views

$(w^2+x^2).(y^2+z^2)$ is always divisible by which of the max no. Where w;x;y;z are positive odd integers?

Q $(w^2+x^2).(y^2+z^2)$ is always divisible by which of the max no where w,x,y,z are positive odd integers? Options given: 20;8;4;2 My Approach: I Choose ($9^2$+$5^2$).($7^2$+$3^2$) to get ...
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1answer
38 views

$3x + 1$ problem: other repetitions [on hold]

The Collatz problem: Pick an integer $x > 0$ If $x$ even: $x = x / 2$, if $x$ odd: $x = 3x + 1$ repeat 2.) as long as you want This algorithm seems to always end up with the loop $4, 2, 1$ My ...
15
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4answers
1k views

Probability that a natural number is a sum of two squares?

Some natural numbers can be expressed as a sum of two squares: $$2=1^2+1^2$$ $$25=3^2+4^2$$ $$50=7^2+1^2$$ If one chooses a random natural number, what would be the probability that that number is a ...
0
votes
2answers
20 views

Simulataneous equations

Suppose you have the following system of linear congruence 2x+5y is congruent to 1 (mod6) x+y is congruent to 5 (mod6) where x,y belong to the set of Integers How would you obtain a general ...
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0answers
37 views

Find all integers $m,n$ for which $m^2+n^2$ is a square and $\sqrt{\frac{2m^2+2}{n^2+1}}$ is rational

This is a repost of my old question here. The question is as follows: Find all integers m and n, such that $m^2 + n^2$ is a square and $\sqrt{\frac{2(m^2+1)}{n^2+1}}$ is rational. I have made no ...
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1answer
21 views

Finding Maximum in a Set of Numbers [on hold]

If I have a set of $n$ numbers: $(a_1,..., a_n)$, then how can I find the two maximum numbers in the set? Suppose that all the numbers are positive integers.
2
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1answer
47 views

Can different tetrations have the same value?

Suppose, we have two numbers $a\uparrow \uparrow b$ and $c\uparrow \uparrow d$. To avoid trivial cases, suppose $a,b,c,d>2$ and $(a,b)\ne (c,d)$. Is there a quartupel $(a,b,c,d)$ with ...
6
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2answers
85 views

Does the Riemann-Hypothesis imply the Twin-Prime-Conjecture?

The Riemann hypothesis (https://en.wikipedia.org/wiki/Riemann_hypothesis) is one of the most important conjectures in number theory. I read that the Riemann hypothesis implies the Goldbach Conjecture ...
5
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1answer
54 views

Proving that and how $ \frac{1}{n}\sum\limits_{p\le n}\lfloor n/p \rfloor - \sum\limits_{p\le n} 1/p $ approaches $0$

Let $p$ denote a generic prime number. By Mertens' second theorem, the sequence $$\sum\limits_{\ p \le n} \frac1p - \log\log n$$ converges to the Meissel-Mertens constant $M\approx 0.2614972$. Now let ...
3
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0answers
39 views

Is there anything known about the zeros of $\displaystyle \sum_{n=1}^{\infty} \left(\frac{1}{\rho_n^s} +\frac{1}{\overline{\rho_n}^s}\right)$?

Assuming the RH and $\rho_n =\frac12 + \gamma_n i$ being the n-th non-trivial zero of $\zeta(s)$, then numerical evidence suggests that: $$f(s) :=\displaystyle \sum_{n=1}^{\infty} ...
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1answer
27 views

Remainder of a number when divisible by a particular number. [duplicate]

What is the remainder when $2^{1990}$ is divided by $1990$?
3
votes
1answer
74 views

Which constellations of primes recur forever?

Having derived much joy and learning from the answers I have received to four previous questions, let me ask one more. Let a constellation of primes be a set of primes that stand in certain fixed ...
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0answers
28 views

Are there other known continued fractions that show the digits of the golden ratio?

I found a few. {16; 5, 1, 1, 5, 22} {161; 1, 4, 11, 1, 1, 3, 6, 1, 13, 8, 1, 6, 1, 4, 1, 1, 2, 1, 1, 1, 1, 13, 2, 1, 3, 8, 1, 2, 19, 1, 54, 1, 19, 2, 1, 8, 3, 1, 2, 13, 1, 1, 1, 1, 2, 1, 1, 4, 1, 6, ...
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1answer
47 views

Crude verification of Goldbach conjecture [on hold]

So the Goldbach conjecture says 'Every even integer greater than 2 can be written as sum of two primes'. Here is what I have roughly done to verify it, using probability. I don't say it is correct but ...
0
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0answers
29 views

How can there be an infinite number of sequentially composite Fibonacci(p)?

I ran into this counting function a(n)>=a(k)+1 for the number of distinct prime factors of the n-th Fibonacci number, at OEIS. Thank you Robert Israel! Thank you for writing the proof there. I had ...
1
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1answer
22 views

Proving a simple modulo equality

I'm probably lacking some basic concept here but I'm trying to prove that $$ ((a \mod k) \cdot k + b) \mod k = (a \cdot k + b) \mod k$$ I get stuck at the passage where, applying distributive ...
0
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2answers
43 views

Can absolute value functions be moved like this?

If I have an expression that looks like $|x-a_1| + |x-a_2| + |x-a_3| + ... + |x-a_n|$ Is it the same as doing $|nx - \sum_{i=1}^{n}a_i|$
0
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1answer
29 views

Calculate the number of zeros in square of ((4404 with base 17)) . ..?

Q Calculate the number of zeros in square of (4404 with base 17)? My approach: @Edit is it right? (4404 at base 17)*(4404 at base 17)=(10G0GF0G at base 17) So, the number of zeros will be 3.
0
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1answer
27 views

Show that (c,a)=(c,b)

In my book I have the implication: If $gcd(a,b)=1$ and $c|(a+b)$, then $gcd(c,a)=gcd(c,b)=1$. It gives me a hint that begins by supposing that $gcd(a,c)=gcd(b,c)=d$. But in my opinion, I do not ...
1
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0answers
28 views

Ways Of Finding Primes and If they are efficient

I am currently in middle school and love number theory. I try and do a proof every day and today I was working on a relatively simple one involving primes. I proved that every prime above 5 can be ...
1
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1answer
61 views

What is the limit of the sum of “last half” part of harmonic series?

I'm looking for the limit of this sum: $\frac{1}{\left\lceil\frac{n}{2}\right\rceil+1}+\frac{1}{\left\lceil\frac{n}{2}\right\rceil+2}+\frac{1}{\left\lceil\frac{n}{2}\right\rceil+3}+\cdots+\frac{1}{n}$ ...
3
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0answers
35 views

Can the determinant of an integer matrix with $k$ given rows be the gcd of the determinants of the $k\times k$ minors of those rows?

I'm interested if the following is true: Let $n\geq k\geq1$ be integers, let $A\in\mathbb Z^{k\times n}$ and denote the $\binom nk$ $k\times k$ minors of $A$ by $A_1,\ldots,A_N$. Then the ...
0
votes
1answer
45 views

how to solve 192-2a^2-a=m(6a+1)?

how to solve $192-2a^2-a=m(6a+1)$ ? or written as $(192-2a^2-a) \equiv 0$ (mod $6a+1$) how to calculate the integer values of $a < 41$ ? thanks to understand that serving: ...
2
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2answers
50 views

Is the sequence $\{0,2,6,12,20,30,…,n(n+1)\}$ admissible for every natural $n$?

Look here : https://en.wikipedia.org/wiki/Prime_k-tuple for the definition of an admissible sequence. I wonder if the sequence of differences of primes can be $\{0,2,4,6,8,...,2n\}$ for every ...
1
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1answer
32 views

raising elements of profinite groups to $p$-adic powers

Let $\widehat{F_2}$ be the profinite free group of rank 2, and let $\widehat{\mathbb{Z}}$ be the profinite completion of $\mathbb{Z}$, and $\widehat{\mathbb{Z}}^\times$ its group of units. For ...
5
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1answer
86 views

Thue equation $ x^4 - 6 x^3 y - x^2 y^2 + 6 x y^3 - y^4=-1 $

I need help solving the Thue equation $ x^4 - 6 x^3 y - x^2 y^2 + 6 x y^3 - y^4=-1 $. It can be written as $ x(x-y)(x+y)(x-6y) = (y-1)(y+1)( y^2 +1) $. From this I found 8 solutions ...
3
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1answer
29 views

Algebraic integers of $\mathbb{Q}(\sqrt{m})$ for $m$ a squarefree integer

I'm currently reading Marcus' "Number Fields," and I'm having difficulty proving the following result: Corollary 2.2: Let $m$ be a squarefree integer. The set of algebraic integers in the quadratic ...
3
votes
2answers
63 views

Prove that for every natural number $n > 2$ there is a prime number between $n$ and $n!$

So I have already read this page with the solution: For all $n>2$ there exists a prime number between $n$ and $ n!$ Now I was able to reason that $p < n!$ Because I was given the hint that ...
2
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1answer
46 views

Numbers relative to their sum of Divisors

Define the D-Ratio as the ratio of a natural number $n$ as: the sum of $n$'s Divisors, excluding 1 and $n$ divided by $n$ itself. [Thus the D-Ratio of $24$ is $$\frac{2 + 3 + 4 + 6 + 8 + 12}{24} = ...
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1answer
30 views

Two Vertical Lines

What does two single vertical lines mean in math. I am thoroughly confused by this question: What describes |3/1|? Use all that apply.
1
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1answer
38 views

Integral of polynomial related to prime divisors

Given the following integral $I_{m,n}=\int_{0}^{1}(1-x^n)^m \mathrm{d}x$. Prove that for any fixed $n$ and for any $m$ $I_{m,n}$ is a rational number and when written in the form $\frac{p}{q}$ with ...
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votes
3answers
56 views

Prove that: 1. $gcd(a,b)=lcm(a,b)$ iff $|a|=|b|$ 2. $k>0\implies lcm(ka,kb)=k lcm(a,bk)$ 3. $a\mid m, b\mid m$, then $lcm(a,b)\mid m$

Let $a,b$ any non-zero integers. Prove that: $gcd(a,b)=lcm(a,b)$ If and only if $|a|=|b|$. If $k>0$, then $lcm(ka,kb)=k lcm(a,bk)$ if $m$ is multiple of $a$ and $b$, then $lcm(a,b)$ divides $m$ ...
0
votes
2answers
58 views

Prove that $\mathbb{Z}[i]$ consists precisely of the elements of $\mathbb{Q}(i)$ which satisfy $x^2 + ax + b=0$, $a,b \in \mathbb{Z}$

I was reading Neurkich's "Algebraic Number Theory" and there was a proof in it that makes no sense. Proposition 1.5: $\mathbb{Z}[i]$ consists precisely of the elements of the extension field ...
5
votes
1answer
54 views

Missing values of the ratio $\frac{(x+y+z)^2}{x^2+y^2+z^2}$

Let $x,y,z$ be some positive integers. Is it true that we cannot find any positive integer $n$ for which $$ \frac{(x+y+z)^2}{x^2+y^2+z^2}=1+\frac{2}{3n}\,\,? $$
3
votes
2answers
78 views

If $x^{100}$ is 31 digit number Then $x^{1000}$ contains how many digits.

If $x^{100}$ is 31 digit number Then $x^{1000}$ contains how many digits. Our Approach: $10^1$ has $2$ digits = $10$ $10^2$ has $3$ digits = $100$ $10^3$ has $4$ digits = $1000$ $10^{30}$ has ...
0
votes
0answers
31 views

Find the value of $b(100)-b(97)-b(96)$ if $b(0)=1$ with given conditions

If for any series $b(n)=2b(n-1)$ when $b(n)$ is odd number and $b(n)=b(n-1)$ if $b(n)$ is even number. then Find the value of $b(100)-b(97)-b(96)$ if $b(0)=1$ Our Approach: I could not ...
2
votes
0answers
26 views

What will be the last number of the set B in which a set B={$2$,$3$,$5$,$6$,$7$,$10$,_ ,_ ,_______} contain $300$ nos.

A set B={$2$,$3$,$5$,$6$,$7$,$10$,_ ,_ ,_______} contain $300$ nos. in which squares and cube of the no. are eliminated. then what will be the last number of the set B? Our Approach: As we have ...