<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<HTML><HEAD>
<META http-equiv=Content-Type content="text/html; charset=iso-2022-jp">
<META content="MSHTML 6.00.2900.2180" name=GENERATOR>
<STYLE></STYLE>
</HEAD>
<BODY bgColor=#ffffff>
<DIV><FONT face="MS UI Gothic" size=2> Hi,
Seqfans.</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic" size=2> The following formula
is well known for proving "Existence of infinite primes".</FONT></DIV>
<DIV><FONT face="MS UI Gothic"
size=2> Product_{1<=i<=n}
p_i + 1</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2> I think that it is
elegant and the easiest. </FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2> But I suppose that
Seqfans might like the most complicated one.</FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2> So, I submit this
sequence.</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> %I
A000001<BR> %S A000001 3, 175, 2336191, 26093310174834487
<BR> %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) *
(Sum_{1<=i<=n} (1/p_i*Product_{1<=k<=n} p_k) )
<BR>
Where p_i means i-th prime.<BR> %C A000001 This is a "Proof of
existence of infinite primes"
sequence.<BR>
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) * (Sum_{1<=i<=n}
(1/p_i*Product_{1<=k<=n} p_k) ) . 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=0, Mod
p_i).<BR> <BR> %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)
*
(p_2*p_3+p_1*p_3+p_1*p_2)<BR>
=
(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_2*p_3+p_1*p_3+p_1*p_2)<BR>
= 31*11*13*17*31
<BR> <BR> %Y A000001 A111392
<BR> %K A000001 none<BR> %O A000001
1,1<BR> %A A000001 Yasutsohi Kohmoto <A
href="mailto:zbi74583@boat.zero.ad.jp">zbi74583@boat.zero.ad.jp</A> </FONT></DIV>
<DIV><FONT face="MS UI Gothic" size=2>
</DIV></FONT></BODY></HTML>