# Tagged Questions

43 views

### Prime factorization of 2^(16) - 1

Trying to show the the prime factorization of $$2^{16}-1$$ without a calculator. I know that $2^{16} - 1$ yields the prime numbers$$3*5*17*257$$ because I calculated $2^{16}-1$ on my calculator ...
29 views

### Let $p \geq 5$ and prime. Show $p^2 + 2$ is divisible by three.

I know I have to use the division algorithm to put into the form $p^2 + 2 = 3q + r$ but everything I've tried after that has lead me to a dead end. I've mainly been trying to show $r=0$ or to make the ...
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### Divergence of a series containing primes

Is there an easy proof showing that the series $1/p$, where $p$ changes over prime numbers, is divergent?
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Given two numbers $a,b \in \mathbb{Z}$, how do we prove that the $p$-adic number of $\gcd(a,b)$ is the same as the minimum for the $p$-adic number of $a$ and the $p$-adic number of $b$? Does this ...
48 views

### Foundational proof for Mersenne primes

I know how to prove that, if $2^n-1$ is prime and $n>1$, then $n$ is prime. But how do we prove that, if $a^n-1$ is prime and $n>1$, then $a$ must equal 2?
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### Fermat number factor probability

I found a question I couldn't solve: What is the probability that $2^{2^{12}}+1$ has a prime factor of $70$ digits? I found this problem hard as many number has more small prime factors than large ...
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### Asymptotic sum of the squares of the first n primes [closed]

I know there is an asymptotic formula for the sum of the squares of the first n primes, but I have been unable to find it.
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### Could you give a definition of what is a superior highly composite number using only words?

I know very well what is a superior highly composite number, but I would like to see how could we (roughly) define what is a superior highly composite number using only words (using no equations and ...
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### Between $n$ and $2n$ there is always a prime number. [duplicate]

Between $n$ and $2n$ there is always a prime number. I was thinking of this and looked it up on the google to find that this is true. Now, I am wondering what is the proof for it? Does any ...