7
votes
5answers
665 views

Induction hypothesis misunderstanding and the fundamental theorem of arithmetic.

The fundamental theorem of arithmetic is made of two parts: The existence part: There exist primes such that for any natural number $j$, we can write $j$ as a product of prime numbers. The ...
2
votes
1answer
33 views

Prove that there are an infinity of prime $ak+b$, $a$ and $b$ coprimes

We have to integers $a,b$. I need to show that if $a$ and $b$ are coprimes then the set of prime numbers of kind $ak+b$ is infinite. How could I show it ? I know how to do that for $4k+3$ or $4k+1$, ...
0
votes
2answers
54 views

Does $\frac{8k-1}{4}$ belongs to $\mathbb{Z}$?

Does $\frac{8k-1}{4}$ belongs to $\mathbb{Z}$ for some $k\in \mathbb{Z}$ ? or we can prove that this never belongs to $\mathbb{Z}$ ?
8
votes
1answer
58 views

Existence of root of a polynomial over $\mathbb F_p$.

I came accross the following question and I can't find an easy proof of this fact : Let $p\geq 17$ be a prime number such that $p\equiv 1 \pmod 4$. Show that for any $z\in \mathbb ...
0
votes
1answer
50 views

Explain theorem in Number theory

can some one explain with a clear example this theorem for me, Let ($A_1$, $A_2$, $A_3$,..., $A_n$) be integars and $p$ a prime number. if $p|(A_1A_2A_3...A_n)$ then there exist some $1 \leq k \leq ...
1
vote
1answer
139 views

Primes created by “n + digital-root(n)” sequences

I've looked at the sequences created by repeatedly adding the digital root of a number to the number until it becomes prime. This is the pseudo-code for the program I've used:   n = 0 ...
3
votes
1answer
156 views

Why is Euler's totient function equal to (p-1)(q-1) when N=pq and p and q are prime? [duplicate]

Why is Euler's totient function equal to $(p-1)(q-1)$ when $N=pq$ and $p$ and $q$ are prime? I had my own proof for it but I really don't like it (it feels not intuitive at all because it requires ...
4
votes
4answers
227 views

By definition, how is a prime number represented?

Even numbers can be easily represented as $2n$. Odd numbers as $2n+1$. An exactly divisible operation can be defined as $n = dq$. But, is there an specific way of representing a prime number, ...
1
vote
1answer
71 views

Solutions for $p$ where $2 p^2 - 1 = q^2$

Consider this equation: $$2 p^2 - 1 = q^2$$ where $p$ and $q$ are prime. After vigorous checking, I couldn't find any solutions $p>29.$ Is it so that $p=5,\;p = 29$ are the only solutions? ...
0
votes
1answer
118 views

Smallest Mersenne prime with 100 million digits?

As some of you are probably aware, the Great Internet Mersenne Prime Search (GIMPS) is managing the search for the largest Mersenne primes of the form $M_p=2^p-1$, where $p$ is itself prime (GIMPS ...
3
votes
1answer
115 views

Efficiency in factoring lists of consecutive numbers

Suppose I'm looking at prime factorizations of numbers in the vicinity of this one: $$ 1354 = 2 \times 677 $$ The smallest prime appears here, and the next prime after that does not. Going one step ...
8
votes
2answers
172 views

Numbers of the form $(p_{1}^{\alpha_{_{_1}}})^{2}+(p_{2}^{\alpha_{_{_2}}})^{2}+\cdots+(p_{n}^{\alpha_{_{_n}}})^{2}=(p_{m}^{\alpha_{_{_m}}})^{2}$

I'm looking for numbers of the form $$(p_{1}^{\alpha_{_{_1}}})^{2}+(p_{2}^{\alpha_{_{_2}}})^{2}+\cdots+(p_{n}^{\alpha_{_{_n}}})^{2}=(p_{m}^{\alpha_{_{_m}}})^{2}$$ where $p_{i}$ are prime numbers, ...
19
votes
5answers
879 views

a big number that is obviously prime?

I once heard it asserted that $91$ is the smallest composite number that is not obviously composite. The reasoning was that any composite number divisible by $2$, $3$, or $5$ is obviously composite, ...
3
votes
2answers
805 views

Can somebody simply explain Wilson's theorem (for a 13 year old)

I am Rohan Kapur. This is my first time posting on the Mathematics site, although I am quite active on StackOverflow, the programming site. I am doing a Islamic Maths assignment at the moment for ...
12
votes
2answers
692 views

nth powers modulo all primes

Let $a \in \mathbb{Z}$, $n \in \mathbb{N}^*$ be integers, and set $P=X^n - a$. Let us consider the three following statements : 1) $P$ has a root in $\mathbb{Z}$ (i.e. $a$ is an nth power) 2) $P$ ...
2
votes
3answers
2k views

How to find next prime number?

By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we can see that the 6th prime is 13. What is the 10001st prime number?