Proving (a weaker form of) Bertrand's postulate

[stevenytc]

It is known that for every n > 1 there is always at least one prime p such that n < p < 2n.

The Bertrand's theorem's proof I read is quite involved. I've thought of another way to prove it but I think it's too simple to be correct. Can anyone please point out my mistakes?

below, Fundamental Theorem means the Fundamental Theorem of Arithmetic

Proof:
#
1) for any n > 1, n can be expressed as a product of prime factors (Fundamental Theorem)
2) Let PF[n] be a set containing all prime numbers that appear in the prime factorization of n.
e.g. PF[6] = {2,3}
3) obviously, if x is in PF[n] then x ≤ n.
4) for all prime numbers that are smaller than n, we look for one that is not in PF[n], if it exists, let it be M.

for M exists:
5a) consider K = n + M, K ≤ 2n (from 4)
6a) now, K is not divisible by any prime smaller than n, since for any prime that divides n, it does not divide M, and M does not divide n. (from 4)
7a) but K has a prime factorization (Fundamental Theorem)
8a) hence there must be a prime in the range [n, 2n]

for M not exists:
5b) consider K' = n+1, K ≤ 2n (assumption, n>1)
6b) now, K is not divisible by any prime smaller than n, since for any prime that divides n, it does not divide 1.
7b) but K has a prime factorization (Fundamental Theorem)
8b) hence there must be a prime in the range [n, 2n]

This proof seems alright to me, but after all I'm just an amateur. I would be grateful if anyone could point out the insufficiency of this proof so that I can learn from my mistakes.

Thanks.

 

[micromass]

It is known that for every n > 1 there is always at least one prime p such that n < p < 2n.

The Bertrand's theorem's proof I read is quite involved. I've thought of another way to prove it but I think it's too simple to be correct. Can anyone please point out my mistakes?

below, Fundamental Theorem means the Fundamental Theorem of Arithmetic

Proof:
#
1) for any n > 1, n can be expressed as a product of prime factors (Fundamental Theorem)
2) Let PF[n] be a set containing all prime numbers that appear in the prime factorization of n.
e.g. PF[6] = {2,3}
3) obviously, if x is in PF[n] then x ≤ n.
4) for all prime numbers that are smaller than n, we look for one that is not in PF[n], if it exists, let it be M.

I think it's not hard to show that M will always exist. For n somewhat large.

for M exists:
5a) consider K = n + M, K ≤ 2n (from 4)

Problem: What if K=2n?? Then you indeed found a prime in [n,2n], but a prime that equals n. Bertrand's postulate says that it must be STRICTLY larger than n!

6a) now, K is not divisible by any prime smaller than n, since for any prime that divides n, it does not divide M, and M does not divide n. (from 4)

This is not true. Take n=8, take M=7. Then K=15 is divisible by 5 and 3 which are both smaller than n.

 

[stevenytc]

ok, I understand now, thanks!

 

[TheFurryGoat]

I think it's not hard to show that M will always exist. For n somewhat large.

For any $k\geq 2$ choose $n$ to be the product of all prime numbers $\leq k.$ Then $\{p\in\mathbb{P} : p < n, p \not |n \}=\emptyset.$ So $M$ need not exist, if I'm not mistaken.

 

[TheFurryGoat]

for M exists:
5a) consider K = n + M, K ≤ 2n (from 4)
6a) now, K is not divisible by any prime smaller than n, since for any prime that divides n, it does not divide M, and M does not divide n. (from 4)
7a) but K has a prime factorization (Fundamental Theorem)
8a) hence there must be a prime in the range [n, 2n]
.

The problem here is, that you use the fact that if $p|(n+M) \text{ and } p|n \Rightarrow p|M.$ But this doesn't cover all the possibilities, namely if $p\not|n \text{ and } p\not|M \text{ but } p|(n+M),$ as micromass pointed out.

 

[micromass]

For any $k\geq 2$ choose $n$ to be the product of all prime numbers $\leq k.$ Then $\{p\in\mathbb{P} : p < n, p \not |n \}=\emptyset.$ So $M$ need not exist, if I'm not mistaken.

Not true. Take k=10. Then n=2*3*5*7=210. There are a lot of primes smaller than 210.

 

[TheFurryGoat]

Not true. Take k=10. Then n=2*3*5*7=210. There are a lot of primes smaller than 210.

woops, sorry, got mixed up with my thoughts, you're right.

 

[coolul007]

Erdos has a very elegant proof of this theorem.