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<DIV><FONT face="MS UI Gothic" size=2> Hi,
Seqfans</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2> I made two proofs for
"Existence of infinite primes".<BR></FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic" size=2> I considered the
following formulas.<BR> 1. N_1 =
Product_{1<=i<n} (Product_{1<=k<=i} p_k +
Product_{i<k<=n} p_k) <BR>
2. N_2 = Sum_{1<=i<=n} (1/p_i*Product_{1<=k<=n} p_k)
<BR>
[Proof1]<BR> If finite primes
p_i exist, 1<=i<=n.<BR> If
N_1 is prime, then for all i,
not(N_1=p_i).<BR> Because, for
all i, p_i<N_1.<BR> If N_1 is
composite, then it must have a prime p which is different from primes
p_i.<BR> Because, for all i,
not(N_1=0, Mod p_i) . <BR>
[Proof2]<BR> Rewrite N_1 to
N_2.<BR> Then the same thing as
Proof1.</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic" size=2> I came across two
</FONT><FONT face="MS UI Gothic" size=2>sequences related these
formulas.</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic" size=2> %I A000001
<BR> %S A000001 2, 5, 28, 162319,<BR> %N
A000001 a(n) = Product_{1<=i<n} (Product_{1<=k<=i} p_k +
Product_{i<k<=n} p_k) <BR> %C A000001 This is
"Proof of existence of infinite primes"
sequence.<BR>
<BR> [Proof
]<BR> Let denote N
Product_{1<=i<n} (Product_{1<=k<=i} p_k +
Product_{i<k<=n} p_k) <BR>
If finite primes p_i exist,
1<=i<=n.<BR> If N is
prime, then for all i,
not(N=p_i).<BR> Because, for all
i, p_i<N.<BR> If N is
composite, then it must have a prime p which is different from primes
p_i.<BR> Because, for all i,
not(N_1=0, Mod p_i) . <BR></FONT><FONT face="MS UI Gothic"
size=2></FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic" size=2> %Y A000001
A024451 <BR> %K A000001 none<BR> %O
A000001 1,1<BR> %A A000001 Yasutsohi Kohmoto <A
href="">zbi74583@boat.zero.ad,jp</A> </FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic" size=2> %S A000002 1, 5, 31,
247, </FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic" size=2> But A000002 already
exists on OEIS as A024451.</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2><BR> </DIV></FONT>
<DIV><FONT face="MS UI Gothic" size=2>
Neil.<BR> Should I comment about the proof on %C line of
A024451? </FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2> </FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2> Yasutoshi</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic" size=2> PS</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2> Are they well known
proofs?</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2> I haven't seen
them.</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2>
</FONT></DIV></FONT></DIV></BODY></HTML>