Composite

Numbers p^q or p^q ±r, where p,q are distinct primes?

February 21st, 2010 | by admin |

What properties do the numbers p^q
or the numbers p^q ±r, where p,q are distinct primes, have?
Do these things have a name?
(r could be prime or composite.)
p^q is obviously composite; p^q ±r might or might not be prime

The subcase p=2 is pretty well studied, and in particular
the subcase p=2, q=some prime, r=-1 is Mersenne (at least, for the definition which requires the exponent q to be prime)
Folks don’t be afraid of it, it won’t bite, it’s just an idle question, I don’t even know whether there is an answer, let alone one anyone cares about.

p^q is composite and probably boring, unless it has some unlikely properties I’m not aware of other than:
- it obeys Fermat’s little theorem (see also: Carmichael numbers)
- it has (q+1) distinct factors (boring)
- it is a ‘Powerful number’, since q≥2
- it is a divisor of some Achilles number (Any interesting conjectures about p^q and Achilles numbers?)

The numbers p^q ±r are maybe more interesting, and might throw up some prime-generating or other formulas (not necessarily good ones, but hey), other than Mersenne’s formula. I’ve only ever seen p=2 or rarely p=3 used, are there any formulas involving higher primes?
- see e.g. the formulas for even and odd perfect numbers.
- see also Wieferich prime
- related but different: Pierpont primes; Wagstaff primes;
Ramanujan-Nagell equation (
p=2, q not required to be prime, r=-7)
For more inspiration, see "Lists of the first prime numbers of various types"
http://en.wikipedia.org/wiki/List_of_special_classes_of_prime_numbers
David – there’s no "right answer", it’s just an amateur question, I’m disappointed people didn’t just have a go at it.

For example, are 5^q ±r, 7^q ±r …
(for r even)
any use as prime-generating formulae?
If not, why is 2 special?

And so on.

Seems to me you would be more likely to get an answer if you ask on sci.math (Usenet or Google Groups).