## amriju 3 years ago Can any of u provide a logical proof of the statement " There is no end to the number of prime numbers?"

1. Eulie

hi @amriju. how are you?

2. amriju

I think I am fine...

3. mikaa_toxica13

there is no endd to numbers period. it goes 999,999,999,999,999,999,999,999,999,999,999,999,999---- and so on. there is no end. so the prime numbers go on and on. no nd to either :) hope i helpd.

4. Eulie

there is no end to the number of prime numbers because numbers go on and on and not all of our numbers are disible by more than just one and itself.

5. amriju

No dear..I need something logical...u kno..give me a theoritical proof...not based on exprience. @mikaa_toxica13

6. mikaa_toxica13

then i am sorry. maybe @Eulie can help :) good luck

7. amriju

how can u prove that...if i wud say after a nuber x..all numbers are divisible...by smaller numbers other than 1...how wud u prove me wrong...? @ Eulie

8. Eulie

Euclid's Proof So let's suppose that there are only finitely many primes, say n of them. We could then make a list of these n primes, say p1, p2, ..., pn. Then consider the integer obtained by multiplying all of the primes, and then adding 1. This new number m is either prime or it is not. But m is clearly larger than all of p1, p2, ..., pn so it cannot be one of the primes. So m is not prime, and therefore must be evenly divisible by some prime. But we are supposing that the only primes are p1 through pn--and none of these primes divides m evenly, for when m is divided by one of these primes, the remainder is 1. This contradiction shows that the supposition that p1, p2, ..., pn is a complete list of all primes must be false. Our conclusion is that there are not finitely many primes; there are infinitely many. Can We Generate all Primes?

9. Eulie

How Many Primes are There? We will now consider the question of how many prime numbers there are. The following investigation of this question is over two thousand years old, and is attributed to Euclid. Two possible answers to the question are that there are either finitely many primes, or there are infinitely many of them. We will show that the first answer is not possible, and therefore there are infinitely many primes. (This approach is referred to as "proof by contradiction." In general, proof by contradiction works like this: we know that there are only two possible answers to a question; we assume that the first answer is the correct one, and then discover that this leads to an inconsistency; and so we conclude that it is actually the second of the two answers that is correct.)

10. estudier

Suppose that p1,p2,p3.....pn are all of the primes. Then consider N = p1*p2*p3*......pn + 1 N is divisible by some prime pk and pk also divides p1*p2*p3..*pn so pk divides N - p1*p2*p3....pn. That is, pk divides 1. Since pk is >=2, contradiction so the original supposition is wrong and there must be another prime not in the list. Evidently u can repeat this argument as often as u please.

11. Eulie

go to http://www.jcu.edu/math/vignettes/primes.htm...... that may help

12. Eulie

@estudier has a point

13. amriju

U are wonderful....both ur display pic( lol) and ur answer....thnx a lot. @Eulie and u have a final point in the proof...@estudier ...i recon i give the former a medal..and the former gives the latter a medal...:P

14. amriju

lol .. i rquested u give estudier a medal...@Eulie

15. Eulie

how eo i do that?

16. amriju

cant do that in this ques anyway....u need to click on best response...and u can do that only once for a ques...where n what do u study anyway..?

17. Eulie

im a senior

18. amriju

does that mean u r not studing any more...? u must be frm the US or Canada....i dont get ur terms like middle school..etc etc...

19. Eulie

im from the us

20. amriju

thats why u use terms like senior....in india here....we wud say a pass out...if u have already got a degree..

21. Eulie

oh im sorry....um i dont know what kindof name that is and u dont hav a pic... ar u male or female?

22. amriju

thats not my real name of crse...and i am a male...or else i wud nt hav chatted wit u for so long.. to be frank...lol

23. Eulie

oh okay... u should put ur pic up so i can see if your as cute as u sound

24. amriju

correctn " as u write"...u havn't hrd my voice ...hey i cud jst catch u later coz i need to go offline now..my mail id...riju_ian@yahoo.com...u cud send urs here if u do not have any probs byezzzz...take care...