What are the Missing Members in this Prime Series?
what is the Significance of this Sequence?
$19$, $17$, $13$, $?$, $?$, $3$, $2$
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1 Answer
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3
The Missing Members may be
$7, 5$ as supposed by Marin Mersenne.
If this is a countdown of
$2, 3, 5, 7, 13, 17, 19, 31, ...$
These are Mersenne exponents: primes $p$ such that $2^p - 1$ is prime. Note $2^{11} − 1 = 2047 = 23 × 89$.
This is OEIS sequence A000043 and currently 51 are known. The exponents listed by Mersenne were: 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257. His list replicated the known primes of his time with exponents up to 19. His next entry, 31, was correct, but the list was incorrect, as Mersenne mistakenly included M67 and M257 (which are composite) and omitted M61, M89, and M107 (which are prime).
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$\begingroup$ @Uvc What is unusual about the missing members? $\endgroup$– hexominoCommented Jun 13, 2019 at 9:26
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$\begingroup$ At first glance, it would have looked like normal prime sequence..11, 7, 5..but because they are Mersenne exponents, 11 is missing. $\endgroup$– UvcCommented Jun 13, 2019 at 9:29