Proving Infinite Primes of 6m+1 and 6m+5 Form Using Euclid's Method

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Homework Help Overview

The problem involves proving that there are infinitely many primes of the form 6m+1 and 6m+5, using a method inspired by Euclid's approach, with a specific modification to the method's application.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the idea of multiplying forms of primes and modifying the result, questioning whether this approach effectively demonstrates the infinitude of the primes in question. There is also consideration of the implications of divisibility and the nature of odd primes.

Discussion Status

The discussion is ongoing, with participants exploring different interpretations of the problem and the methods suggested. Some guidance has been offered regarding the construction of products of primes and the implications of adding or subtracting values from these products.

Contextual Notes

Participants note the constraints of the problem, including the requirement to use a specific method and the implications of the forms of numbers being discussed. There is also a recognition of the limitations of certain approaches in proving the desired outcome.

cragar
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Homework Statement


Except 2 and 3 , prove that their an infinite amount of primes
of the form 6m+1 and 6m+5 for some integer m
It says to use Euclid's method but replace the +1 with a -1.

The Attempt at a Solution


Would I just multiply some of these forms together and subtract 1
(6m+1)...(6n+1)-1=x
If I divide my new number x by (6m+1) it won't divide it evenly.
this doesn't seems like it proves it, am I on the right track?
 
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I wouldn't think so. You would only be able to show that x is not divisible by primes of the form 6m+1. You won't be able to show it is not divisible by other primes.
 
If you multiply k primes of the form (6m+1) and l primes of the form (6m+5), you get a product of the form (6m+1) for even l and (6m+5) for odd l. This allows to find new primes, assuming a finite number of primes of those types. I am not sure how to get an infinite number of (6m+1)-primes, but it is certainly possible for (6m+5)-primes.
 
ok thanks for the replies Ill work on it.
 
Suppose p to be the largest prime of a given one of those two types, and construct the product of all primes (except a certain few) up to p.
 
So I make a product of all primes of the form 6m+1 and then add one
to it. I am still not really sure exactly what you guys are telling me, Ill keep thinking about it.
another thing i noticed is that all of the odd numbers greater than 5 are of the form
6m+1 or 6m+3 or 6m+5, but we know that 6m+3=3(2m+1) so it never produces primes.
so we know because their are an infinite amount of odd primes that
either or both 6m+1 or 6m+3 contain an infinite amount of primes.
 
Last edited:
cragar said:
So I make a product of all primes of the form 6m+1 and then add one
That's not what I wrote: "the product of all primes (except a certain few) up to p."
 
1 is a bad thing to add, as the result will be divisible by 2.
so we know because their are an infinite amount of odd primes that
either or both 6m+1 or 6m+3 contain an infinite amount of primes.
Right
 

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