Thread: Primes in Arithmetic Progression

Dirichelt's Theory that there are infinitely many primes in any arithmetic progresssion. Does anyone know its proof? Nowhere I look it says "The proof of this is beyond the scope of this book" even in Hardy's and Wright's book! Come on, how complicated a proof can be?

Originally Posted by ThePerfectHacker

Dirichelt's Theory that there are infinitely many primes in any arithmetic progresssion. Does anyone know its proof? Nowhere I look it says "The proof of this is beyond the scope of this book" even in Hardy's and Wright's book! Come on, how complicated a proof can be?

In "Number Theory" by W. Narkiewicz it takes about 11 pages. Having
skimmed through the section it does not look too exotic, but that was
a very superficial skim

RonL

Hi every one!

I don't know if the photo joined whith PerfectHacker is recent, or his, but sudanly he looks a litle older, I had imagine him whith some sort of the hair gelly they try to sale young people in TV advertising! But lets keep our subject in mind.

So Dirichlet demonstrate the (probably) folowing théorème in the Ninetheen Century, but he had to do it because it is a very useful thèorème, he probably reach an idea and developed it in a eleven page (we were told) demonstration so he could use it. He probably didn't loose much time searching for a shorter demo.
I have myself think of it a few time because I need it, and at this very moment to fight against anachronisme.

The Théo says that if A and B are prime togethere there is an infinite set of integer K were A+KB is prime.
So we just have to demonstrate that whith any A and B prime together their is almost one (non nul) integer K whith A+KB prime, and the rest folows by induction.
Thats means if we take C for wich CB>A that all the number ending whith an A when write in CB base (GFA=G*(CB^2)+F(CB)+A for exemple) are not prime and of course those ending whith the C-1 "digit" of the form A+LB.
So the mathématicien who is conjecturing such a thing feel very confident
about the thruness(?) of it.
How long would it take(in term of space) you to do it if you cannot reach Dirichlet demo. It's a challenge!
I have myself a litle idea but had not try to formalise it whith my steacky fingers,(had just the idea last night):
Lets put it as a legend picked upon Louise de Montculque diary(sory for my translation) :
<<Erasthophene who was very poor and who was giving math lecon to rich pupils of his time made some litle bets whith his pupils from time to time.
when he was filling his famous crible (which where of CB whidth) he use to bet upon some unmarked number if it would be prime, of course he manage to loose from time to time to keep the game interesting, but he had a trick: when he arrived to a column which were not completly marked ( not a divisor of CB I guess) he knew that the first unmarked case(?) of the column was prime>>
Thats true at least if the ratio between the high of the part of the column who is marked above the considered point of the column and the same stuff under this point does not exeed the number represented by this point.
The rest is only conjecture!
I hope I was not to unclear and that someone would try to test this conjecture whith his brain or a computer.
So good Christmass and Hapy new years , and bests wishes for all of you!(it does not need any penny anyway)
Thanks
Your faithfully Picaziet pablo.

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