0
votes
0answers
40 views

Nondiagonalizable matrix with successive primes?

The 4x4-matrix filled from left to right and from top to bottom with successive primes, starting with the prime $21005627$ , is not diagonizable. Is there a 3x3-matrix with the same property ? I ...
1
vote
4answers
57 views

solutions to linear equations involving prime numbers?

Suppose we have the two equations: $2Z - p = Xq$ $2Z - q = Yp$ where $X,Y,Z \in \mathbb{N} $ and $p,q \in \mathbb{P} - \left\{2\right\} $ Are there any solutions where $Z$ isn't prime?
1
vote
3answers
33 views

Linear Independence for different fields

I have a statement for a space over $R^n$: {x, y, z} is linearly ind. $\implies$ {x + y, x + z, y + z} is linearly independent Quick proof: a(x+y) + b(x+z) + c(y+z) = 0 $\implies$ (a+b)x + ...
0
votes
0answers
36 views

How small can we make a modulus and still perform linear algebra on these pairs?

We can work with numbers of the form $(a^n + a^m)$, where $a$, $n$, and $m$ are all naturals, and $-v \le m \le v$ and $-v \le n \le v$. There is one more possibility: $a^n$ could be replaced by $0$, ...
1
vote
1answer
255 views

Pisano periods of fibonacci mod

The wikipedia article on Pisano periods utilises the Binet's formula and quadratic residues to find $f(n)$ such that $F_n=f(n) \pmod{p}$ where $p$ is a prime number and $F_n$ is a Fibonacci number. ...
4
votes
2answers
144 views

What is the need for classifying numbers like integer, whole number etc?

what are the everyday life examples where we use the classification. I feel all the math behind the scenes(in computers weather etc ) is highly abstracted. I am looking for strong answers to tell the ...
2
votes
2answers
274 views

About linear combinations of primes

$a,b,c$ are natural numbers whose greatest common divisor is $1$. $a,b,c\in\mathbb{N}^*$, $(a,b,c)=1$ Try to write down the expression using $a,b,c$ of the biggest natural number $M$ that cannot be ...
1
vote
1answer
88 views

Dirichlets theorem on primes

I want to use Dirichlets theorem on primes for my diploma thesis. I want to use following form Let $a,b\in\mathbb{N}$, such that $\gcd(a,b)=1$. Then the set $\{a\cdot n+b| n\in\mathbb{N}\}$ contains ...
0
votes
2answers
226 views

Standard Basis of the Finite Field of Prime Numbers

A little info regarding this field: Addition and multiplication in $Z^n_p$ behave as usual but with the remainder taken upon division by $p$. Ex: $Z_3$ will only consist of the three integers ...
0
votes
1answer
48 views

Finding a number that satisfies $q = x_a({p_a}^2)+4$

$$q = x_a({p_a}^2)+4$$ Let $p_n$ be a sequence of consecutive prime numbers starting from 3. ($p_a$ represents a prime number in the sequence, and $x_a$ is the corresponding $x$ to the prime number.) ...
1
vote
2answers
179 views

What is the fastest algorithm to check if a number has only 3 divisors?

Which is the fastest way to check if a number has only 3 unique factors ? Any help will be highly appreciated?
2
votes
2answers
145 views

Matrices with elements that are a distinct set of prime numbers: always invertible?

Inspired by a previous question, given a square non-symmetric matrix whose elements are all prime but distinct from each other, does this guarantee that the matrix is invertible? It's easy to see ...
0
votes
2answers
609 views

How to calculate all the prime factors of lcm of n numbers?

I have n numbers stored in an array a, say a[0],a[1],.....,a[n-1] where each a[i] <= 10^12 and n <100 . Now,I need to find all the prime factors of the LCM of these n numbers i.e., LCM of ...
1
vote
1answer
364 views

Quadratic forms and prime numbers in the sieve of Atkin

I'm studying the theorems used in the paper which explains how the sieve of Atkin works, but I cannot understand a point. For example, in the paper linked above, theorem 6.2 on page 1028 says that if ...
1
vote
0answers
128 views

Binary sequences in primes

Is anything known about these problems? If we make a string S of 0's and 1's with 1 in n'th position if the the nth prime $p_n$ is of the form $1+m 2^{9^{9^{9^{9}}}}$, else 0, does every finite string ...