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<DIV>:In article <<A
href="mailto:Pine.GSO.3.96.980813170545.6488B-100000@atlantis">Pine.GSO.3.96.980813170545.6488B-100000@atlantis</A>>,<BR>:
"don <" <<A
href="mailto:dsmcdona@actrix.gen.nz">dsmcdona@actrix.gen.nz</A>>
wrote:<BR>:<BR>:> > A single partition of 619 proves it is prime
below.<BR>:> ><BR>:> > Integer 619 = 15,249
- 14,630<BR>:>
>
= 3.13.17.23 - 2.5.7.11.19.<BR>:> ><BR>:> > Every prime factor
beginning at 2 up to 23 appears on<BR>:> > one side or the other of this
special [partition] of 619,<BR>:> > but never on both sides at
once.<BR>:> ><BR>:> > Therefore, integer 619 is prime.<BR>:>
> q.e.d.<BR></DIV>
<DIV><FONT size=2>This got me wondering which primes are provable via this
method.</FONT><BR><FONT size=2>We can show p >= 2 prime if we can exhibit a
> 0, b > 0, q prime </FONT><FONT size=2>with</FONT></DIV>
<DIV><FONT size=2></FONT> </DIV>
<DIV><FONT size=2> p = a +- b;</FONT><FONT
size=2> ab = primorial(q);</FONT><FONT
size=2> p < nextPrime(q)^2.</FONT></DIV>
<DIV><FONT size=2></FONT> </DIV>
<DIV><FONT size=2>Here is a list of primes I found</FONT></DIV>
<DIV><FONT size=2></FONT> </DIV>
<DIV><FONT size=2>5 = 3 + 2 < 5^2<BR>7 = 2*5 - 3 < 7^2<BR>11 = 2*3 + 5
< 7^2<BR>11 = 3*7 - 2*5 < 11^2<BR>13 = 2*5 + 3 < 7^2<BR>13 = 3*5 - 2
< 7^2<BR>13 = 5*11 - 2*3*7 < 13^2<BR>17 = 2*7*13 - 3*5*11 < 17^2<BR>17
= 3*5 + 2 < 7^2<BR>23 = 2*3*5 - 7 < 11^2<BR>29 = 3*5 + 2*7 < 11^2<BR>29
= 5*7 - 2*3 < 11^2<BR>31 = 2*3*11 - 5*7 < 13^2<BR>31 = 3*7 + 2*5 <
11^2<BR>37 = 2*3*5 + 7 < 11^2<BR>37 = 2*3*7 - 5 < 11^2<BR>37 = 2*5*7 -
3*11 < 13^2<BR>41 = 3*5*11*19 - 2*7*13*17 < 23^2<BR>41 = 3*5*13 - 2*7*11
< 17^2<BR>41 = 5*7 + 2*3 < 11^2<BR>47 = 2*3*7 + 5 < 11^2<BR>47 = 7*11 -
2*3*5 < 13^2<BR>67 = 2*3*5*7 - 11*13 < 17^2<BR>67 = 2*5*7 - 3 <
11^2<BR>73 = 2*5*7 + 3 < 11^2<BR>83 = 3*5*7 - 2*11 < 13^2<BR>89 = 2*5*11 -
3*7 < 13^2<BR>97 = 5*11 + 2*3*7 < 13^2<BR>101 = 2*3*11 + 5*7 <
13^2<BR>101 = 3*7*11 - 2*5*13 < 17^2<BR>103 = 2*5*7 + 3*11 < 13^2<BR>103 =
3*5*7 - 2 < 11^2<BR>107 = 2*5*7*11 - 3*13*17 < 19^2<BR>107 = 3*5*7 + 2
< 11^2<BR>107 = 7*11 + 2*3*5 < 13^2<BR>127 = 3*5*7 + 2*11 < 13^2<BR>131
= 2*5*11 + 3*7 < 13^2<BR>139 = 2*7*11 - 3*5 < 13^2<BR>151 = 3*5*11 - 2*7
< 13^2<BR>157 = 3*5*7*11*13 - 2*17*19*23 < 29^2<BR>163 = 3*7*13 - 2*5*11
< 17^2<BR>181 = 2*11*13 - 3*5*7 < 17^2<BR>227 = 2*5*17*19 - 3*7*11*13 <
23^2<BR>239 = 2*3*5*11 - 7*13 < 17^2<BR>263 = 2*3*11*13 - 5*7*17 <
19^2<BR>349 = 2*5*7*13 - 3*11*17 < 19^2<BR>389 = 3*5*13*17 - 2*7*11*19 <
23^2<BR>619 = 3*13*17*23 - 2*5*7*11*19 < 29^2<BR>709 = 5*7*19*23 -
2*3*11*13*17 < 29^2<BR></FONT></DIV>
<DIV><FONT size=2>There don't seem to be any more up to q = 59.</FONT></DIV>
<DIV><FONT size=2>I would be surprised if there were any
more.</FONT></DIV></BODY></HTML>