A049535. Feb. 16-20, 5 consecutive dates containing a square.

[David Wilson]

:In article <Pine.GSO.3.96.980813170545.6488B-100000 at atlantis>,
:  "don <" <dsmcdona at actrix.gen.nz> wrote:
:
:> > A single partition of 619 proves it is prime below.
:> >
:> > Integer   619  = 15,249  -  14,630
:> >                = 3.13.17.23 - 2.5.7.11.19.
:> >
:> > Every prime factor beginning at 2 up to 23 appears on
:> > one side or the other of this special [partition] of 619,
:> > but never on both sides at once.
:> >
:> > Therefore, integer 619 is prime.
:> >          q.e.d.

This got me wondering which primes are provable via this method.
We can show p >= 2 prime if we can exhibit a > 0, b > 0, q prime with

    p = a +- b;    ab = primorial(q);    p < nextPrime(q)^2.

Here is a list of primes I found

5 = 3 + 2 < 5^2
7 = 2*5 - 3 < 7^2
11 = 2*3 + 5 < 7^2
11 = 3*7 - 2*5 < 11^2
13 = 2*5 + 3 < 7^2
13 = 3*5 - 2 < 7^2
13 = 5*11 - 2*3*7 < 13^2
17 = 2*7*13 - 3*5*11 < 17^2
17 = 3*5 + 2 < 7^2
23 = 2*3*5 - 7 < 11^2
29 = 3*5 + 2*7 < 11^2
29 = 5*7 - 2*3 < 11^2
31 = 2*3*11 - 5*7 < 13^2
31 = 3*7 + 2*5 < 11^2
37 = 2*3*5 + 7 < 11^2
37 = 2*3*7 - 5 < 11^2
37 = 2*5*7 - 3*11 < 13^2
41 = 3*5*11*19 - 2*7*13*17 < 23^2
41 = 3*5*13 - 2*7*11 < 17^2
41 = 5*7 + 2*3 < 11^2
47 = 2*3*7 + 5 < 11^2
47 = 7*11 - 2*3*5 < 13^2
67 = 2*3*5*7 - 11*13 < 17^2
67 = 2*5*7 - 3 < 11^2
73 = 2*5*7 + 3 < 11^2
83 = 3*5*7 - 2*11 < 13^2
89 = 2*5*11 - 3*7 < 13^2
97 = 5*11 + 2*3*7 < 13^2
101 = 2*3*11 + 5*7 < 13^2
101 = 3*7*11 - 2*5*13 < 17^2
103 = 2*5*7 + 3*11 < 13^2
103 = 3*5*7 - 2 < 11^2
107 = 2*5*7*11 - 3*13*17 < 19^2
107 = 3*5*7 + 2 < 11^2
107 = 7*11 + 2*3*5 < 13^2
127 = 3*5*7 + 2*11 < 13^2
131 = 2*5*11 + 3*7 < 13^2
139 = 2*7*11 - 3*5 < 13^2
151 = 3*5*11 - 2*7 < 13^2
157 = 3*5*7*11*13 - 2*17*19*23 < 29^2
163 = 3*7*13 - 2*5*11 < 17^2
181 = 2*11*13 - 3*5*7 < 17^2
227 = 2*5*17*19 - 3*7*11*13 < 23^2
239 = 2*3*5*11 - 7*13 < 17^2
263 = 2*3*11*13 - 5*7*17 < 19^2
349 = 2*5*7*13 - 3*11*17 < 19^2
389 = 3*5*13*17 - 2*7*11*19 < 23^2
619 = 3*13*17*23 - 2*5*7*11*19 < 29^2
709 = 5*7*19*23 - 2*3*11*13*17 < 29^2

There don't seem to be any more up to q = 59.
I would be surprised if there were any more.
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