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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1answer
167 views

How prove this $|\{n\sqrt{3}\}-\{n\sqrt{2}\}|>\frac{1}{20n^3}$

Prove that $$|\{n\sqrt{3}\}-\{n\sqrt{2}\}|>\dfrac{1}{20n^3}$$ let $t=\{n\sqrt{2}\}-\{n\sqrt{3}\}$ and $k=[n\sqrt{3}]-[n\sqrt{2}]$ then we have ...
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2answers
65 views

Show that if $a$ has order $3\bmod p$ then $a+1$ has order $6\bmod p$.

Show that if $a$ has order $3\bmod p$ then $a+1$ has order $6\bmod p$. I know I am supposed to use primitive roots but I think this is where I am getting caught up. The definition of primitive root ...
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0answers
133 views

Existence of prime pairs

Will there always exist a prime pair of the form (p, p+l) for any l where gcd(p,l) = 1 and l is even? Can we always conjecture that there exist infinitely many prime pairs of the form(p, p+l) when ...
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1answer
126 views

Attempted exercise using Littlewood's theorem

This was an exercise to try to show we can use Littlewood's theorem$^1$ to prove that $$\lim_{N \to \infty}\frac{1}{N}\sum_{n=1}^{N} \frac{g(n)}{\log p_n} = 1 \hspace{30mm}(1)$$ If $\vartheta(p_k) ...
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1answer
159 views

Euler's phi function $\phi(n)$ is even for all $n \geq 3$; when is it not divisible by $4$?

Problem 1: Show that $\phi(n)$ is even for all $n \geq 3$. Proof: Assume that n is a power of 2, let us say that $n=2^k$, with $k \geq 2$. By the Phi Function Formula, we have $\phi(n) = ...
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1answer
91 views

Series equivalent to $\sum p_k$

Looking at a theorem of Chebyshev, I noticed that $$\sum_{n=0}^{\infty} \sum_{p_k < n} \frac{(\log p_k)^n}{n!} = 2 + 3 + ...+ p_k.$$ Proof. Letting $x = \log p_k$ and writing out the expansion of ...
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2answers
161 views

For a positive integer $n$, $p_n$ denotes the product of the digits of n, and $s_n$ denotes the sum of the digits of n.

For a positive integer $n$, $p_n$ denotes the product of the digits of $n$, and $s_n$ denotes the sum of the digits of $n$. What is the number of integers between 10 and 1000 for which $p_n + s_n = n$ ...
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1answer
406 views

About two consecutive integers which are sum of squares

I am looking for all two consecutive integers A and A+1,which can be represented as sums of two squares $A=a^2+b^2$ and $A+1=c^2+d^2$, $a,b,c,d>0$
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262 views

How many primes in the first 6000 primes have a particular property

How many prime numbers are there in the first $6000$ prime numbers that are the quotients of other prime numbers in the following way $(P_1^2-1)/(P_2^2-1)=P_3$ where $P_1$ , $P_2$ and $P_3$ are ...
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1answer
79 views

existence of a prime $p$ for which $a$ is a primitive root

It is known that every prime $p$ has a primitive root modulo $p$. Is every number $a$ which is not a perfect square a primitive root modulo $p$ for some prime $p$? If it is a square, we already have ...
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5answers
120 views

Show $60 \mid (a^4+59)$ if $\gcd(a,30)=1$

If $\gcd(a,30)=1$ then $60 \mid (a^4+59)$. If $\gcd(a,30)=1$ then we would be trying to show $a^4\equiv 1 \mod{60}$ or $(a^2+1)(a+1)(a-1)\equiv 0 \mod{60}$. We know $a$ must be odd and so $(a+1)$ ...
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0answers
146 views

Is zero a cluster point of $n\sin n$?

I have seen somewhere that if $0\le \alpha<1$, then zero is a cluster point of the sequence $n^\alpha \sin n, n=1,2,\cdots$. My question is what if $\alpha=1$? Or $\alpha>1$?
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1answer
72 views

$xy= z^2$ and $x, y$ are individual squares

We know that concatenation of $xy = z^2$ for $x = 4^2 = 16$ and $y = 3^2 = 9$. Here $169 = 13^2 \implies z = 13$. Now my question is how to prove this is the first such set of positive integers $(x, ...
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0answers
77 views

Frobenius endomorphism on supersingular elliptic curve

Let we have supersingular curve $E(\bar{\mathbb{F}_q})$. Is it true that for every point $P$ $q$-Frobenius endomorphism $\pi_q$ can be write as $A + [q]B$ where $P = A + B$? It is true if ...
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2answers
88 views

Prove if $(l,m)=1$ and $l\mid mn$, then $l\mid n$.

I just took my number theory final and this was on the exam as the second question. It said to use the canonical decomposition of $l, m$ and $n$ for the proof. This is what I put down on the exam: ...
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3answers
258 views

Computing large modular numbers

How do you compute large modular arithmetic such as $8^{128}$ $mod$ $100$ or $10^{111}$ $mod$ $137$ or $3^{100}$ mod $17$? I know that one way is repeated squaring. For the first one, my book says 16, ...
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1answer
135 views

Coding Theory - Fourier/Walsh/Hadamard Transform

Hi guys these questions are from my homework. I am not asking you to solve my homework. Instead, i just need some help in getting started. Please list down some steps as I am very lost. Also, if you ...
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4answers
305 views

Find the complete solution to the simultaneous congruence.

I'm having trouble understanding the steps involved to do this question so any step by step reasoning in solving the solution would help me study for my exam. Thanks so much! $$x\equiv 6 ...
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2answers
62 views

How to find $a,b\in\mathbb{N}$ such that $c = \frac{(a+b)(a+b+1)}{2} + b$ for a given $c\in\mathbb{N}$

Suppsoe that $$c = \frac{(a+b)(a+b+1)}{2} + b$$ Now $c$ is given - how does one find satisfying $a, b$?
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1answer
96 views

Largest power of a prime dividing $q^m-1$

For positive integers $x$ and $d$ let $v_d(x)$ be the largest power of $d$ dividing $x$. Let $q>1$, $m$ a natural number, and $l$ a prime dividing $q-1$. Then I want to show that ...
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3answers
186 views

Number theory sums of squares.

Show that if q is a number which can be expressed as the sum of two perfect squares, then 2q and 5q can also be expressed as the sum of two perfect squares.
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94 views

The mathematics behind Sobol sequences

I am using Sobol sequence as random number generator in a computer program. Beyond just making the program work, I would like to learn the mathematics behind the Sobol sequence (and other ...
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2answers
133 views

How many integer solutions of “a” exist for this equation?

$a\times b = 8 \times (a + b)$ I have used Wolfram Alpha and it has given me 14 integer solutions. But how can we find those solutions ? Which method ? edit: integer solutions are as following: $a ...
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1answer
70 views

probability of a number not having factors below n?

I want to know the probability that a number is not divisible by any prime smaller than or equal to n. ie, I find a big number, I trial division it up to 300000 and don't find any factors. I want to ...
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1answer
64 views

Function for unique hash code

I am interested in finding $F(x,y)$, such that $x$ and $y$ $\in \mathbb Z^+$ and $F(x,y)$ is one to one function i.e., $F(x,y)$ is unique for any unique unordered pairs of $x$ and $y$. Regards, ...
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1answer
48 views

How to calculate the index between two complex lattices?

Let $\Lambda$,$\Lambda'$ be two complex lattices and $m\neq 0\in\mathbb{C}$ satisfying $m\Lambda\subset\Lambda'$. Suppose $\omega_1,\omega_2$ are the basis of $\Lambda$, $\omega'_1,\omega'_2$ are the ...
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1answer
1k views

Goldbach Conjecture Consequences

I have been looking into the Goldbach Conjecture pretty recently and I have often heard that it would have far-reaching consequences. However, I haven't found many of the actual consequences. I was ...
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4answers
1k views

How do you know when you square/multiply or just square when performing “Square and Multpily” algorithm?

I'm having difficulty understanding the Square and Multiply algorithm. As you can see below, we are trying to compute Y = x ^d (mod N) where x=4 and d=12, which is 1100 in binary. There is ...
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1answer
321 views

Divergence of the sum of the reciprocals of the primes

I know there are many ways to prove that the sum of the reciprocals of the primes diverges, but does the following argument work? The cardinality of the set of all prime numbers is obviously ...
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3answers
108 views

Two problem number-theory

$p$ and $q$ are prime, $p \neq q , a$ is integer: 1)$p^{q-1}+q^{p-1} \equiv 1 \pmod{pq}$ 2)$p|(a^p + (p-1)!a) $
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445 views

Is there an equation for the sum of alternating cubes?

The following sequences for sum of alternating cubes: Odd cubes: [1, 28, 153, 496, 1225, 2556, 4753, 8128, 13041, 19900, 29161, 41328, 56953, 76636, 101025, 130816, 166753, 209628, 260281, 319600] ...
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2answers
84 views

Function expansion help

I want to write a function, $f(k,a,b)$, I made, in terms of combinations of the fractional part function, $$ j\left\{\frac{c \ }{d}k\right\},$$ where $c,d,$ and $j$ are any integers. The function is ...
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3answers
164 views

Are there programs out there that try to derive linear recurrence given a string of numbers?

Wolfram isn't helping me much so I am curious if there are other programs out there. I don't know what degree it is, but I have a series of numbers and I'd like to determine the linear recurrence ...
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1answer
67 views

How to better understand this function?

let $P(n) = n^4 + an^3 + bn^2 + cn$ $M(a,b,c)$ returns largest $m$ that divides $P(n)$ for all n then let function $S(N)$ return the sum of all $M(a,b,c)$ for $1 \le a,b,c \le N$ I don't need ...
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1answer
58 views

What's the probability to find a value of $t<T$ where $|P_k(it)|<\epsilon$?

Given $P_k$, the truncated Prime $\zeta$ function, defined like $$ P_k(it):=\sum_{n=1}^k p_n^{it}, $$ where $p_n$ is the $n$th prime. What's the probability to find a value or range of $t$ less than ...
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1answer
246 views

How many ways are there of coloring the vertices of a regular $n$-gon

How many ways are there of coloring the vertices of a regular $n$-gon with all $p$ colors ($n,p \ge 2$), such that each vertex is given one color, and every color isn't used for two adjacent vertices? ...
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1answer
186 views

Matrices that satisfy $AB = 0$ and $A^2 = B^2$

I want to make a set of matrices that satisfies all the following: 1) $A^2 = B^2$, $C^2 =D^2$..... where $A,B,C,D...$ are matrices 2) $AB = 0$, $CD = 0$..... 3) All matrices in the set commute. 4) ...
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1answer
384 views

System of 3 linear congruences

Find all solutions: $$\begin{cases} x\equiv 39 \mod(189) \\ x\equiv 25 \mod(539) \\ x\equiv 399 \mod(1089) \end{cases}$$ But $189=3^3\cdot7$, $539=11\cdot7^2$ and $1089=3^2\cdot 11^2$, so I can't ...
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2answers
271 views

Function with a Modular Inverse

For a combinatorics problem I have a function, $h(x)$ that is always divisible by five, but it is calculated in pieces, e.g. $h(1) = 43 + 7$. The final function that I need is $f(x) = (h(x) / 5) ...
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3answers
151 views

Properties of Mediants

If $\frac{a}{c} > \frac{b}{d}$, then the mediant of these two fractions is defined as $\frac{a+b}{c+d}$ and can be shown to lie striclty between the two fractions. My question is can you prove ...
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0answers
60 views

series: can the result be zero for a continuous interval of its argument?

I'm considering the series $$ f_c(x) = \sum_{k=c}^\infty \left( c^{k-1} \binom{k}{c} \cdot \prod_{j=1}^{k-1} (x-1/j) \right) $$ where the parameter $c \in \mathbb N ,c \gt 0$ and fixed for a certain ...
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2answers
163 views

Proof that a sum of the first period of powers of integer roots of rationals is irrational

I'm looking for a simple proof or a reference to any proof that For $j \in ℤ$, $0<j<m$, when each $k^j \notin ℚ$ and each $d_j \in ℚ$, $d_j \neq 0$, then $\sum_{j=1}^{m-1} d_j k^j \notin ℚ$ ...
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0answers
82 views

Binary forms of degree n

Im trying to show that the binary form $x^{n−1}+x^{n−2}yα+x^{n−3}y^2α^2+...+y^{n−1}α^{n−1}$ is bounded below by $ c*y^{n−1}$ where c is some explicit constant. For the case n=3 this is fairly easy, ...
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3answers
547 views

$\mathbb{Q}(\sqrt{d})$ with specific integral basis

I would like some help with the following question. Ireland and Rosen (ch.13#10) For which $d$ does $\mathbb{Q}(\sqrt{d})$ have an integral basis of the form $\alpha, \alpha '$ where $\alpha '$ ...
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1answer
126 views

Producing infinite family of transcendental numbers

Weierstrass proved the result [Lindemann-Weierstrass theorem] that if $a_1, \cdots, a_n$ are reals linearly independent over the rationals, then $e^{a_1}, \cdots, e^{a_n}$ are algebraically ...
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2answers
400 views

twin prime conjecture

Whether I am correct or wrong I don't know. If there are any corrections, please let me know. Let $p_n$ = product of all primes. (of course we can go still beyond as we know $p_n$ is infinite). Now ...
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2answers
313 views

If $\gcd(a,b)=1$ , and $a$ is even and $b$ is odd then $\gcd(2^{a}+1,2^{b}+1)=1$?

How to prove that: $\gcd(a,b)=1 \Rightarrow \gcd(2^{a}+1,2^{b}+1)=1$ ,where $a$ is even and $b$ is odd natural number For example: $\gcd(2^8+1,2^{13}+1)=1 , \gcd(2^{64}+1,2^{73}+1)=1$ I know ...
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1answer
249 views

Fencing the Group size,and its implication to Finiteness of Tate-Shafarevich Group

This question is an interesting one,not like my previous one. Can we judge the size of a Quotient Group by seeing the size of its constituents ? To add something ,Suppose consider a group ...
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3answers
391 views

Patterns in Sequences

I've heard in a movie that for any sequence of numbers, there is a nice formula for generating that sequence. So, for example if I write: 1,2,1,2,3,3,1,2,3,1,2,4,... There is a formula for ...
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3answers
212 views

Diophantine equation

If one solves the Diophantine equation $cx + by = a$; i.e., $cx = a - by = a \pmod{b}$ formally, then the answer is $x = (a/c) - (b/c)y$, but the integer character and information is lost and not ...