If sum of two positive integers $a$ and $b$ is a prime then $a$ and $b$ are mutually coprime.
When the sum is a prime
It should be trivial but I just found it today
If sum of two positive integers $a$ and $b$ is a prime then $a$ and $b$ are mutually coprime.
If sum of two positive integers $a$ and $b$ is a prime then $a$ and $b$ are mutually coprime.
"Je le vois, mais je ne le crois pas!" - Georg Ferdinand Ludwig Philipp Cantor
Re: When the sum is a prime
Woow.. trivial but i never noticed that. thank you 
Every logical solution to a problem has its own beauty.
(Important: Please make sure that you have read about the Rules, Posting Permissions and Forum Language)
(Important: Please make sure that you have read about the Rules, Posting Permissions and Forum Language)
Re: When the sum is a prime
let $p$ be the sum of $a$ and $b$ and $a$ and $b$ arent coprime so
$a+b=p$
$k(a_1+b_1)=p$ but this contradicts with the definition of prime numbers so $a$ and $b$ has to be co-prime
$a+b=p$
$k(a_1+b_1)=p$ but this contradicts with the definition of prime numbers so $a$ and $b$ has to be co-prime
Re: When the sum is a prime
But this simple observation has become a very serious fact.
Dirichlet's theorem says that if $gcd(a,b)=1$ then the sequence $an+b$ contains infinitely many primes.Surprisingly this is easy to sense but too hard to prove,over hundreds of pages.
Dirichlet's theorem says that if $gcd(a,b)=1$ then the sequence $an+b$ contains infinitely many primes.Surprisingly this is easy to sense but too hard to prove,over hundreds of pages.
One one thing is neutral in the universe, that is $0$.
Re: When the sum is a prime
hmm...so the opposite version of the proposed problem is similar to Dirichlet's theorem. (However, $(a,b)=1$ does not guarantee that $an+b=p\; \forall \; n \in \mathbb{N}$ )
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin
Please install LaTeX fonts in your PC for better looking equations,
learn how to write equations, and don't forget to read Forum Guide and Rules.
Please install LaTeX fonts in your PC for better looking equations,
learn how to write equations, and don't forget to read Forum Guide and Rules.
Re: When the sum is a prime
I didn't say that,just infinitely many
One one thing is neutral in the universe, that is $0$.