Veryprimes, Quiteprimes

[Antreas P. Hatzipolakis]

FWD MESSAGE -----------------------------------------------------------------

Subject: Re: Veryprimes defined
From: Jim Ferry <jferry at uiuc.edu>
Date: Thu, 09 Sep 1999 10:45:20 -0500
Newsgroups: sci.math

Clive Tooth wrote:
>
> Informally, a number is a veryprime if it is a long way from being divisible
> by its trial factors.
>
> More formally: a positive integer, V, is a veryprime iff:
>
> * V is prime.
>     and
> * For every odd prime, p, which is less than the square root of V, we have:
>   V = (p-1)/2 (mod p) or V = (p+1)/2 (mod p).

Being somewhat looser about "long way from divisible" doesn't add many
more veryprimes.  Consider these three versions of the definition:

An integer V > 1 is veryprime iff all primes p <= sqrt(V) satisfy:

|2 (V mod p) - p| <= 1        (very strong)
|2 (V mod p) - p| <= sqrt(p)  (strong)
|2 (V mod p) - p| <= p/2      (weak)

Pardon the computery use of "V mod p" (0 <= (V mod p) < p).

Then the (original) very strong veryprimes are
2, 3, 5, 7, 11, 13, 17, 19, 23, 37, 43, 47, 53, 67, 73, 137.

The strong veryprimes are the above together with
227.

The weak veryprimes are (both of) the above together with
103, 107, 157, 173, 347, 487, 773.

(I checked the first 100,000 primes in each case.)

Being much looser produces a more interesting definition:

An integer V > 1 is *quiteprime* iff all primes p <= sqrt(V) satisfy:

|2 (V mod p) - p| <= p + 1 - sqrt(p).

Of the first 10 primes, 10 are quiteprime; of the first 100, 73; and
so on, as indicated here:

10/10, 73/100, 256/1000, 504/10000, 584/100000

Anyone care to comment (rigorously or heuristically) on the
finiteness and/or asymptotic density of overprimes?
                                        ^^^^^^^^^^
                                  read: quiteprimes (aph)


|             Jim Ferry              | Center for Simulation  |
+------------------------------------+  of Advanced Rockets   |
| http://www.uiuc.edu/ph/www/jferry/ +------------------------+
|    jferry@[delete_this]uiuc.edu    | University of Illinois |

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APH