619 = 3.13.17.23 - 2.5.7.11.19

[Don Reble]

> 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

I think you should allow products of empty sets of primes,
that is, the number 1. This gives a few more one-line proofs:
    3 = 2 + 1
    5 = 3*2 - 1
    7 = 3*2 + 1
    29 = 5*3*2 - 1
    31 = 5*3*2 + 1
(But only 3's primality was in doubt. :-)

---

In fact, so long as each prime up to the square-root appears on exactly
one side, the one-line proof is valid. There can be repeated primes, and
other primes.

     2 = 3 - 1
     3 = 2*2 - 1
     5 = 3*2 - 1
     7 = 2*2*2 - 1
    11 = 2*7 - 3
    13 = 2*2*2*2 - 3
    17 = 2*2*5 - 3
    19 = 2*11 - 3
    23 = 2*13 - 3
    29 = 5*7 - 2*3
    31 = 2*2*3*3 - 5
    37 = 2*2*2*5 - 3
    41 = 2*5*5 - 3*3
    43 = 2*2*2*2*3 - 5

So we have this number sequence:

    1 1 1 1 3 3 3 3 3 6 5 3 9 5 3 7 21 9 3 49 35 5 7 21 15 25 30 28
    21 7 203 100 28 15 126 14 63 35 253 520 910 105 264 665 1155 165
    504 1155 858 156 495 91 539 715 198 507 550 275 143 720 627 2002
    2618 5695 4692 7735 2100 5460 4515 561 2860 1071 6783 49742
    35321 53482 2926 74613 62244 45780 40755 4199 46189 53907 43605
    61047 58786 48620 4180 62985 29640 24871 16530 38779 33150 25432
    36465 14014 51480 840684 187473 482328 448305 123786 361284
    229653 1061760 102102 665665 984929 274703 301392 287385 14630
    490314 326235 156332 534888 141372 426075 424764 447005 111435
    464607 374255 1385670 14586 735471 1229865 357170 213486 621775
    858429 20748 1311550 391391 114342 304980 798798 488631 134849
    204930 1025202 255255 81396 1425586 12785102 3624478 9023651
    5719467 8193218 27698440 5427540 909568 11492663 6734455 1283975
    1057485 3312023 4511749 8786778 5861310 108263805 1343034
    63740820 51571135 44931029 214937415 62805735 29567097 71348116

a(n) = the smallest positive number which furnishes a "one-line proof"
       for primality of the p(n), the n'th prime. A one-line proof looks
       like this:
	    101 = 2*3*3*7 - 5*5
       For each prime Q up to the square-root of p(n), either the left
       product or the right product is divisible by Q, but not both. It
       follows that the difference is not divisible by any such Q, and
       so is prime. The sequence gives the right (smaller) number.

The numbers get big, since the product is at least the primorial, and is
usually bigger. For the 600th prime, it's
    4409 = 3*3*3*13*19*23*31*61*367*9504211
	 - 2*2*5*7*11*17*29*37*41*43*47*53*59*139 (1011724380916715620)

--
Don Reble       djr at nk.ca