Question

Prove that there exist infinitely many primes of the form 6n + 5.

Answer 1

so like a think try proving it by mathinduction

so for n=1

6*1+5=11 is prime

rewrite for n=k

6k +5 suppose it prime and try prove it for k=k+1

for k=k+1 will get

6(k+1)+5=6k+6+5 =6k+11

so now check it for k=1

6*1 +11 =6+11 =17 what is prime too

hope these will help you

Answer 2

jhonyy9 Thank you for your reply

Answer 3

Induction may work to show it is a prime.

But how do we prove the "infinite" number of primes?

Answer 4

@dumbcow

Answer 5

@phi

Answer 6

@thomaster

Answer 7

@amistre64

Answer 8

@ParthKohli

Answer 9

well, most proof of infinite primes deal with contradiction

Answer 10

assume 6n+ 1 is the largest prime of this type for some maximum value of n

6n+1 = p

and determine that there some k>n that is a prime seems to be how some proffs go that i can recall

Answer 11

Thank you for your reply @amistre64

Answer 12

Your approach seems similar to https://math.dartmouth.edu/archive/m25f10/public_html/homework/m25hw3sol.pdf

Answer 13

But how do you get to 6n+1?

Answer 14

Why not, say, 6n+7?

Answer 15

6n+5, seems mistyped that part lol

Answer 16

its asking for the specifics of 6n+5, which is just an even number and an odd number added like 2n+1, all primes but 2 are odd so there is most likely a subset of primes within the set of 6n+5

Answer 17

but yes, that link looks similar to what i was considering as an approach to the solution

Answer 18

Hmm so do you mean 2n+1 is a subset of 6n+5?

Answer 19

well, 2n+1 is all odd numbers, so 6n+5 is missing some odds so it would be the subset

Answer 20

6n+5 = p, for the largest prime of some maximal n (therefore n and its associated prime are finite)

n = (p - 5)/6 , which leads into n being a mod of 6 which seems to play a part in proffing it

Answer 21

i wonder, since p is the largest prime, but n is defined for all primes ... can we conclude that primes themselves are infinite and that n would therefore not have a maximal value?

Answer 22

i spose n has to be n integer so that wouldnt be sufficient

Answer 23

most likely the link you posted has a better run thru that im giving :) since its along the same idea

Answer 24

Thank you for your answers:)
Just wondering though, how did the author in the link get to "N must be a product of only primes of the form 6k + 1"?
Is this some offshoot from remainder theorem?

Answer 25

hmm, seems like the maximal N is being defined by its own prime factorization

Answer 26

6k+5 = N is wht they are using for notation

let N be a prime number defined as 6.p1.p2....pn -1

from which they start to make the conclusions