# proof sequence

Thu Nov 10 03:52:27 CET 2005

```    Hi, Seqfans
I made two proofs for "Existence of infinite primes".

I considered the following formulas.
1. N_1 = Product_{1<=i<n}  (Product_{1<=k<=i} p_k +
Product_{i<k<=n} p_k)
2. N_2 = Sum_{1<=i<=n}  (1/p_i*Product_{1<=k<=n} p_k)
[Proof1]
If finite primes p_i exist, 1<=i<=n.
If N_1 is prime, then for all i, not(N_1=p_i).
Because, for all i, p_i<N_1.
If N_1 is composite, then it must have a prime p which is different
from primes p_i.
Because, for all i, not(N_1=0, Mod p_i) .
[Proof2]
Rewrite N_1 to N_2.
Then the same thing as Proof1.

I came across two  sequences related these formulas.

%I A000001
%S A000001 2, 5, 28, 162319,
%N A000001 a(n) = Product_{1<=i<n}  (Product_{1<=k<=i} p_k +
Product_{i<k<=n} p_k)
%C A000001 This is "Proof of existence of infinite primes" sequence.

[Proof ]
Let denote N  Product_{1<=i<n}  (Product_{1<=k<=i} p_k +
Product_{i<k<=n} p_k)
If finite primes p_i exist, 1<=i<=n.
If N is prime, then for all i, not(N=p_i).
Because, for all i, p_i<N.
If N is composite, then it must have a prime p which is different
from primes p_i.
Because, for all i, not(N_1=0, Mod p_i) .

%Y A000001 A024451
%K A000001 none
%O A000001 1,1
%A A000001 Yasutsohi Kohmoto   zbi74583 at boat.zero.ad,jp

%S A000002 1, 5, 31, 247,

But A000002 already exists on OEIS as A024451.

Neil.
Should I comment about the proof on %C line of A024451?

Yasutoshi

PS
Are they well known proofs?
I haven't seen them.

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