I've been toying with the expression p^n + 2^m as a prime generator,

where p is a prime and n and m are from 1 to p.

Based on the little data I have looked at (below), I would hazard a

guess that for every odd p, p^n + 2^m produces at least one prime.

Perhaps at least 8 primes. (With more data I would like to put

tighter restraints on that guess.) Also, I would guess that over the

range of odd p's, p^n + 2^m will generate any and every prime at

least once.

Here is the data. After this it gets a little more interesting, so

bear with this. Also, since I did this manually there may be an error

somewhere.

3^1 + 2^1 = 5

3^1 + 2^2 = 7

3^1 + 2^3 = 11

3^2 + 2^1 = 11

3^2 + 2^2 = 13

3^2 + 2^3 = 17

3^3 + 2^1 = 29

3^3 + 2^2 = 31

(8 primes generated)

5^1 + 2^1 = 7

5^1 + 2^3 = 13

5^1 + 2^5 = 37

5^2 + 2^2 = 29

5^2 + 2^4 = 41

5^3 + 2^1 = 127

5^3 + 2^5 = 157

5^4 + 2^4 = 641

(8 primes generated)

7^1 + 2^2 = 11

7^1 + 2^4 = 23

7^1 + 2^6 = 71

7^2 + 2^2 = 53

7^2 + 2^6 = 113

7^3 + 2^2 = 347

7^3 + 2^4 = 359

7^4 + 2^4 = 2417

7^5 + 2^2 = 16811

7^5 + 2^4 = 16823

7^5 + 2^6 = 16871

7^7 + 2^2 = 823547

(12 primes generated)

11^1 + 2^1 = 13

11^1 + 2^3 = 19

11^1 + 2^5 = 43

11^1 + 2^7 = 139

11^1 + 2^9 = 523

11^2 + 2^4 = 137

11^3 + 2^7 = 1459

11^4 + 2^4 = 14657

11^4 + 2^8 = 14897

11^5 + 2^1 = 161053

11^5 + 2^3 = 161059

11^5 + 2^9 = 161563

11^7 + 2^9 = 19487683

11^7 + 2^11 = 19489219

11^9 + 2^1 = 2357947693

11^9 + 2^5 = 2357947723

(16 primes generated)

13^1 + 2^2 = 17

13^1 + 2^4 = 29

13^1 + 2^6 = 79

13^1 + 2^8 = 269

13^2 + 2^2 = 173

13^2 + 2^6 = 233

13^2 + 2^10 = 1193

13^3 + 2^4 = 2213

13^3 + 2^10 = 3221

13^5 + 2^8 = 371549

13^6 + 2^2 = 4826813

13^8 + 2^8 = 815730977

13^10 + 2^2 = 137858491853

13^13 + 2^4 = 302875106592269

(14 primes generated)

Surely it is just a coinicidence that in all 5 cases the number of

primes generated is even?

Now, if we restrict the exponents m and n so that each are of the

form 2^c, (c can be zero) we get

3^1 + 2^1 = 5

3^1 + 2^2 = 7

3^2 + 2^1 = 11

3^2 + 2^2 = 13

(4 primes generated)

5^1 + 2^1 = 7

5^2 + 2^2 = 29

5^2 + 2^4 = 41

5^4 + 2^4 = 641

(4 primes generated)

7^1 + 2^2 = 11

7^1 + 2^4 = 23

7^2 + 2^2 = 53

7^4 + 2^4 = 2417

(4 primes generated)

11^1 + 2^1 = 13

11^2 + 2^4 = 137

11^4 + 2^4 = 14657

11^4 + 2^8 = 14897

(4 primes generated)

13^1 + 2^2 = 17

13^1 + 2^4 = 29

13^1 + 2^8 = 269

13^2 + 2^2 = 173

13^8 + 2^8 = 815730977

(5 primes generated)

Now I add a few more primes, but my computer's calculator only goes

so high and then looses digit precision, so each is incomplete.

17^1 + 2^1 = 19

17^2 + 2^2 = 293

17^4 + 2^4 = 83537

17^4 + 2^8 = 83777

17^4 + 2^16 = 149057

17^8 + 2^4 = 6975757457

17^8 + 2^16 = 6975822977

17^16 ???

19^1 + 2^2 = 23

19^2 + 2^8 = 617

19^4 + 2^4 = 130337

19^8 + 2^16 = 16983628577

19^16 ???

23^4 + 2^4 = 279857

23^4 + 2^8 = 280097

23^16 ???

29^1 + 2^1 = 31

29^2 + 2^8 = 1097

29^2 + 2^16 = 66377

29^8 ???

29^16???

It is tempting to further guess that there is at least one prime

generated for each of these. But what I found very surprising was

this: if the two exponents add up to an even number greater than 7

then the generated prime seems to always ends in a seven! I haven't

yet investigated why this might be. Maybe it's just a temporary blip

but it is interesting.

Mark