Number Theory Help: Proving Expression is Equal to Prime Number

[Little Gem]

Hi guys, i m just a begineer in number theory.
While solving some questions ,i came across a doubt.

The expression: a+bx
here, gcd(a,b)=1
There always exists a value of x(where x is a integer) such that the above
expression is equal to a prime number.

Can anyone prove the above statement (if it is true).

Also,please suggest some good book on number theory for begineers.
Thanks in advance.

 

[Gib Z]

a+bx=p
x(a/x +b)=p
a/x + b=p/x.

P is prime be definition, so p/x can not be an integer.

a/x + b can not be an integer.
a/x can not be an integer.

Lets let this non integer equal t.
a/x=t
a=xt
Since a has to be an integer, and t is not an integer, x can not be an integer either.

This theorem is false in the natural numbers, or the integers.

 

[Little Gem]

Thanx for solution,
But I have a doubt in :
a/x=t
a=xt
Since a has to be an integer, and t is not an integer, x can not be an integer either.

Here ,how can we say that product of a rational number and integer is not a
integer. As,here x is not a prime(assume) may be factor of denominator of t.

Also,a+bx=p
x=(p-a)/b
means;
p=a(mod b)

Here,just studying a special case,
a=1 and b=r(r is a prime)
1+rx=p (r,p are both primes)
rx=p-1
x=(p-1)/r
p=1(mod r)
This equation has integra solutions.
Above is derived from a
Statement that for any prime p there exists a complete residue system modulo n,all whose members are primes.
So, a can be anything from 1,2,3,...r-1
and r,p are primes.

 

[Gib Z]

Umm I may have gone onto this question intuitively rather than a solid proof.

The product of a rational integer and a non integer can only be another integers if the non integer contains a factor in its denominator. So we have to prove that t is of the form C/nx. But, if it is, the a=c/n. so T =a/x. Since that is true, it is of the form c/nx and that makes my proof wrong >.< but hopefully that leads you in a good direction. Sorry about the mistake

 

[Dick]

Do you have any reason to believe that that is an 'easy' problem? There is a proof by Dirichlet that a series like you have defined contains an infinite number of primes. But the proof uses complex analysis and isn't elementary. One might hope that the job of proving the series contains at least one prime might be easier than showing it contains an infinite number. But I'm really not sure it is.

 

[Little Gem]

Well Dick,you may be right that my question is not that easy,but as i told that
myself being just a begineer ,i may not have duly reconginsed the depth in the question.
But,my intution says that simple expressions like 1+px(p being a prime)
will definitely yield a prime.But,had no idea that even simple problem like that are not that simple at all.

Well,can you please tell from where can i find that proof by Dirichlet.

 

[Dick]

Here's a pdf I found:

modular.fas.harvard.edu/129/projects/weissman/project.pdf

You might also try to find the alternative proof by Selberg cited in the paper - which doesn't use complex numbers but is "long and not particularly enlightening".

 

[Dick]

Also the newsgroup sci.math is a great place to post questions like this. There are some smart people there.

 

[AlephZero]

a+bx=p
x(a/x +b)=p
a/x + b=p/x.

P is prime be definition, so p/x can not be an integer.

That's the first mistake in your proof. p/x can be an integer, if x = 1.

Work through an example like 2 + 3.5 = 17 to find more mistakes.