Mersenne primes with P=1 mod 4 and P=3 mod 4 - Page 2

GP2 · 2004-01-14, 21:30

A Usenet posting from 20 years ago shows that Martin Gardner back in the 1960s was interested in the "infuriating hints of order" here (except he approached it from the perspective of perfect numbers ending in the digits 6 or 8.

http://groups.google.ca/groups?hl=en...1%40dciem.UUCP

only_human · 2004-01-15, 00:27

(cringing) Bearing in mind all said above about Wagstaff's conjecture et. al., could
p 1 mod 4
and
p 3 mod 4
(although not leading digits) be influenced by Benford's Law?
http://www.utm.edu/research/primes/n...nfordsLaw.html

(edited: I expect NO! )

GP2 · 2018-08-06, 16:40

As of today, for the known Mersenne primes 2^P−1 there are 30 exponents P with P=1 mod 4 and 19 exponents P with P=3 mod 4 (and obviously 1 with P=2 mod 4).

So in the past (nearly) fifteen years, the discrepancy has diverged a bit more.

As mentioned earlier, some theoretical considerations for a discrepancy in frequency between the two cases of 1 mod 4 and 3 mod 4 are summarized at Chris Caldwell's page, with a citation to the original article, although the theory doesn't really match the magnitude of the observed discrepancy.

Interestingly, for the 43 currently known Wagstaff primes (2^P+1) /3, there is the opposite situation. There are 15 exponents P with P=1 mod 4 and 28 exponents P with P=3 mod 4.

2004-01-15, 00:27	#13
only_human "Gang aft agley" Sep 2002 2×1,877 Posts	(cringing) Bearing in mind all said above about Wagstaff's conjecture et. al., could p 1 mod 4 and p 3 mod 4 (although not leading digits) be influenced by Benford's Law? http://www.utm.edu/research/primes/n...nfordsLaw.html (edited: I expect NO! ) Last fiddled with by only_human on 2004-01-15 at 00:37

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2004-01-14, 21:30	#12
GP2 Sep 2003 5×11×47 Posts	A Usenet posting from 20 years ago shows that Martin Gardner back in the 1960s was interested in the "infuriating hints of order" here (except he approached it from the perspective of perfect numbers ending in the digits 6 or 8. http://groups.google.ca/groups?hl=en...1%40dciem.UUCP

2018-08-06, 16:40	#14
GP2 Sep 2003 A19₁₆ Posts	As of today, for the known Mersenne primes 2^P−1 there are 30 exponents P with P=1 mod 4 and 19 exponents P with P=3 mod 4 (and obviously 1 with P=2 mod 4). So in the past (nearly) fifteen years, the discrepancy has diverged a bit more. As mentioned earlier, some theoretical considerations for a discrepancy in frequency between the two cases of 1 mod 4 and 3 mod 4 are summarized at Chris Caldwell's page, with a citation to the original article, although the theory doesn't really match the magnitude of the observed discrepancy. Interestingly, for the 43 currently known Wagstaff primes (2^P+1) /3, there is the opposite situation. There are 15 exponents P with P=1 mod 4 and 28 exponents P with P=3 mod 4.