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• Hi all 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
Message 1 of 1 , Jul 2, 2003
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Hi all

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
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