# another proof sequence

Wed Nov 30 08:45:26 CET 2005

```     Neil wrote :
>The OEIS will be on holiday for the rest of the year!
>But only very important new sequences and Comments should be submitted.

I don't know if it is important or not.
If not, ignore it.
I will post it again next year.

%I A000001
%S A000001 3, 35, 75361
%N A000001  a(n) = (Product_{0<=e_i<=1}  (Product_{1<=i<=n} p_i^e_i +
Product_{1<=i<=n} p_i^(1-e_i)))^(1/2)
Where p_i means i-th prime.
%C A000001 This is a "Proof of existence of infinite primes" sequence.
Proof. Let N = (Product_{0<=e_i<=1}
(Product_{1<=i<=n} p_i^e_i + Product_{1<=i<=n} p_i^(1-e_i)))^(1/2) . Suppose
there are only a finite number of primes p_i, 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 divisor p which is different from primes p_i. Because,
for all i, not (N_1=0, Mod p_i).

%e A000001 a(3)=
((1+p_1*p_2*p_3)*(p_3+p_1*p_2)*(p_2+p_1*p_3)*(p_2*p_3+p_1)*(p_1+p_2*p_3)*(p_1*p_3+p_2)*(p_1*p_2+p_3)*(p_1*p_2*p_3+1))^(1/2)
=
(1+p_1*p_2*p_3)*(p_3+p_1*p_2)*(p_2+p_1*p_3)*(p_2*p_3+p_1)
=        31*11*13*17

%Y A000001 A111392
%K A000001 none
%O A000001 1,1
%A A000001 Yasutsohi Kohmoto   zbi74583 at boat.zero.ad.jp

I am not sure if the representation on %N line is correct.
Is it understandable?

Yasutoshi

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