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## 20013Re: primes of the form (x+1)^p-x^p

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• Apr 7, 2009
--- In primenumbers@yahoogroups.com, "Mike Oakes" <mikeoakes2@...> wrote:

>
> My own record (and favourite PRP of all, being so "Mersenne-like") is
> 3^336353-2^336353
> at 160482 digits.
>

Very cool. To notch it up on the coolness factor one could express
the prime exponent 336353 as

3*(2^(2^3+3^2) - 3^(3^2) + 3^(2*3)) - 1

:)

On a not too related note, here's something that may be of interest.

Consider primes generated by 2^m + 3^n and 2^n + 3^m , with m < n.

I've looked at n from 1 to 1000, and there is always at least
one m < n such that 2^m + 3^n OR 2^n + 3^m is prime. But it comes
close to failing on many occasions. (Why am I always attracted to
this kind of exploration?)

Here are the outcomes which came close to failing, where there were
3 or fewer m that resulted in a prime.

Format: (n, number of m less than n such that 2^m+3^n or 2^n+3^m is prime)

(1,1) (2,2) (3,2) (4,3) (5,3) (6,3) (9,3) (11,3) (25,2)
(33,3) (34,3) (54,1) (69,2) (70,3) (97,3) (103,3) (115,3)
(117,2) (118,3) (120,3) (121,3) (122,2) (129,1) (131,2)
(135,1) (139,3) (150,3) (157,2) (161,3) (166,3) (170,1)
(175,1) (185,1) (190,3) (194,3) (200,2) (201,2) (206,3)
(211,3) (213,3) (218,3) (236,2) (240,2) (242,3) (266,3)
(274,1) (280,1) (285,3) (293,3) (294,3) (321,2) (322,3)
(324,3) (335,1) (338,3) (348,2) (351,3) (376,2) (383,3)
(397,3) (398,3) (405,3) (407,2) (415,2) (420,3) (422,3)
(435,3) (445,2) (455,2) (459,2) (460,1) (473,3) (489,1)
(493,3) (506,2) (507,3) (543,2) (547,1) (549,2) (555,3)
(562,2) (565,3) (566,3) (567,2) (570,2) (572,3) (580,2)
(586,3) (591,2) (603,3) (609,2) (611,1) (614,1) (627,3)
(641,3) (651,3) (685,2) (700,3) (711,3) (717,1) (721,3)
(729,1) (736,3) (745,3) (746,1) (747,3) (751,3) (770,3)
(775,3) (798,2) (811,3) (813,1) (819,1) (821,1) (826,1)
(830,3) (835,3) (845,3) (851,1) (859,3) (869,3) (879,3)
(884,3) (887,2) (899,3) (901,3) (911,3) (913,2) (943,2)
(951,3) (958,3) (966,1) (970,3)

The rate of close calls seem to decrease, but slowly. But what
surprised me initially was this: I expected that composite n would be
more represented than they are for close calls.

For instance let n have a factor of 3. Immediately one third of the
m's less than n are disqualified from generating a prime, because if
m contains a factor of 3 then both 2^n+3^m and 2^m+3^n will contain
a factor of 2^3+3^3 = 35.

But as it is, of the 133 n's listed above, only 45 have a factor of
3. About what one would expect if having a factor of 3 made no
difference. What about n with factors of 5? 33 of the 133 n's have a
factor of 5, so the composite effect might be having some effect
there, not sure.

Mark
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