proof sequence

[kohmoto]

    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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