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Old 2021-09-26, 01:25   #24
Dr Sardonicus
 
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Feb 2017
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With a = 33*2^2939064 - 5606879602425*2^1290000 - 1 and d = 33*2^2939063 - 5606879602425*2^1290000, a == 2 (mod 5) and d == 4 (mod 5), so a + 2*d is divisible by 5. Thus a, a + d, a + 2d is not a 3-term AP of primes.

5606879602425*2^1290000 - 1 No prime factors < 2^28.
33*2^2939063 - 1 prime (table lookup)
33*2^2939064 - 5606879602425*2^1290000 - 1
99*2^2939063 - 5606879602425*2^1290001 - 1
33*2^2939065 - 16820638807275*2^1290000 - 1 Divisible by 5

Estimates of base-ten logs of a - d, a, a + d:

(log(33)+2939063*log(2))/log(10) 884747.64066010744143595512237352714687
(log(33)+2939064*log(2))/log(10) 884747.94169010310541715033611242187136 (Matches value given here.)
(log(99) + 2939063*log(2))/log(10) 884748.11778136216109839241740143040198

Thus a - d, a, a + d is (presumably) a 3-term AP of primes. Luckily, a - d is an 884748 decimal digit number as well as a, so assuming this is the first term of an AP-3 the digit count is still good.

If a - 2d = 5606879602425*2^1290000 - 1 happens to be prime, then a - 2d, a - d, a, a + d is a 4-term AP.

But a - 2d "only" has 388342 decimal digits (base-ten log is 388341.44312776625375912068666134298687, approximately)
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