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Old 2019-01-09, 03:34   #254
LaurV
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Most probably, Mark 1 could handle 512 bits numbers with its memory and registers (which is 2^9, quite convenient power of 2). Then everything makes sense, as they stopped just before 521.

@GP2, half way by the number of primes (like in 51/2), or by total number of digits?
(like a string, 2357131719316189107127......)

Because, if it is the first, it won't be hard, the most of the MPs are the "small" ones, hehe, I already wrote 12 of them in the string above, and not yet to a quarter of the string, by far...


Edit: hey, look, 512 and 51/2 only differ by a slash... We should ask that enzo guy, maybe it means something....

Last fiddled with by LaurV on 2019-01-09 at 03:39
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Old 2019-01-09, 06:22   #255
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Quote:
Originally Posted by LaurV View Post
@GP2, half way by the number of primes (like in 51/2), or by total number of digits?
(like a string, 2357131719316189107127......)
139 out of 259 digits in the string. So a bit more than half, actually.

I use the memory method described a few pages ago. For instance, 6972593 is "sheep kennel bomb". A kennel full of sheep instead of dogs is blown up by a terrorist. Vivid and outrageous imagery strung together in a narrative. You could also think of alternatives, like "cheap Queen album".
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Old 2019-01-09, 08:55   #256
LaurV
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I also memorized many of them (in no special order) by just... memorizing the numbers (which sound some peculiar rhythm in my head, or in my mouth). Your method sounds very interesting, but I don't understand how to transform those words into numbers fast in my head without helping tools (like phone keypad, or whatever). I may be too old for it, when I think "telephone", what pops to my head is a device with a rotating disk with holes...

Edit: but I have to recognize that the analogy did here with the computer keypad (mid row, two up, two down) for M51 put its exponent instantly in my head, and most probably I will never forget it.

Last fiddled with by LaurV on 2019-01-09 at 08:59
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Old 2019-01-09, 22:20   #257
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Quote:
Originally Posted by GP2 View Post
That can't be right. Your numbers are off. The Mersenne primes in that section are 521 and 607.

I'm halfway to memorizing the complete list of Mersenne prime exponents. It's the new digits-of-pi.
My mistake, they tested up through 509 and just missed the next prime at 521, thanks!
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Old 2019-01-10, 13:14   #258
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Quote:
Originally Posted by GP2 View Post
I use the memory method described a few pages ago.
The "mnemonic major system?" I looked it up. It gave letters or sounds associated (somehow) with each digit. OK, I might be able to remember those if I drilled for a week or two. Perhaps not.

I move on to the examples. The "one digit peg" example nouns for the digits 1 to 3 all begin with the letter h. Let's see here. The letter h is unassigned. OK, the example word for 0 is hose, and the letter s is associated with 0. So, I also have to remember which letters are unassigned, ignore them, and look for assigned letters?

Geez, let's see how I do trying to recall the first few Mersenne prime exponents I was just looking at in the last few posts for a few minutes. OK, I had the first 6 locked in already.

2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607.

I can remember a few others; 3127 (or is it 3217, I'll have to check), 8191, 11213, 82589933. There are some between 607 and 3127 (3217?) and a bunch of larger ones.

Oh, wait. 56^2 = 3136, so 3127 = 53*59. So the exponent can't be 3127...
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Old 2019-01-10, 14:29   #259
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1279 is just 127 with a 9 appended. 2203 is pretty easy to memorize as is 44497.
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Old 2019-01-10, 15:43   #260
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Quote:
Originally Posted by paulunderwood View Post
1279 is just 127 with a 9 appended. 2203 is pretty easy to memorize as is 44497.
Hmm. 1279 is also 1729 with the middle two digits transposed. And, of course, 1729 is the taxicab number in the anecdote about Ramanujan. It is also the smallest Carmichael number whose prime factors form an AP.

OK, then 2203 and 2281 both begin with 22. And next is 3217. So let's see.

2, 3, 5, 7, 13, 17, 19, 31, 61, 89,107,127, 521, 607, 1279, 2203, 2281, 3217

Did I remember them all? (count count count) That's 18, check. Hooray! I'm through the 1950's!

Hmm. Dates (or at least decades) of discovery are another handle. I can probably memorize how many were discovered each decade since the 1950's. That will provide a check. Interesting...
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Old 2019-01-11, 05:47   #261
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1279 and 11213 were the first exponents we learned at "over 3 digits", the first because we spent a lot of time (years!) trying in and out to factor M1061 and M1277, and the last due to the nice story behind (the one with the stamp).



Where did you get 8191 from? (I am sure MM13 is not prime, since we are hunting for its factors on ET's site).
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Old 2019-01-11, 22:19   #262
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philmoore is right the Manchester Mark 1 was used to search for Mersenne primes (however did not find any new ones) before the SWAC was used for such a search (and found the five next Mersenne primes). So I was incorrect.

Speaking about M521, it is one rare example of a prime that is both Mersenne and Woodall, since \[2^{521}-1 = 512\cdot 2^{512}-1\] I wonder if there are any larger examples of that. Whenever an exponent is of form \(p=2^k+k\), then that \(M(2^k+k)\) is also a \(W(2^k)\).

/JeppeSN

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Old 2019-01-11, 23:52   #263
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Quote:
Originally Posted by JeppeSN View Post
Speaking about M521, it is one rare example of a prime that is both Mersenne and Woodall, since \[2^{521}-1 = 512\cdot 2^{512}-1\] I wonder if there are any larger examples of that. Whenever an exponent is of form \(p=2^k+k\), then that \(M(2^k+k)\) is also a \(W(2^k)\).
Never mind larger examples, there's no smaller example.

The only other p=2^k+k which is a Mersenne prime exponent is k=1, p=3, but then W(k) = 1.

The next larger 2^k+k which are prime are for k=15, p=32783 and k=39, p=549755813927 and then k=75, 81, 89, 317, 701, 735, 1311, 1881, 3201, 3225... (and no others less than 10,000), and the corresponding p are astronomical.

I think we can formulate some kind of "New Woodall Conjecture" with little risk of finding a counterexample.

Last fiddled with by GP2 on 2019-01-11 at 23:54
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Old 2019-01-12, 00:18   #264
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Quote:
Originally Posted by LaurV View Post
Where did you get 8191 from? (I am sure MM13 is not prime, since we are hunting for its factors on ET's site).
I got it from my mind slipping a cog. Lucky I'm doing my rote memorization from a real-deal list
:-D

Actually, the fact rattling around in my head about this exponent was, 8191 is the first Mersenne prime that is not a Mersenne prime exponent.

I had read this (I think) in a Martin Gardner "Mathematical Games" column in the 1960's, about the "useless elegance of perfect numbers."

[Google Google, toil and trouble] M8191 was proven composite in 1954, so this was known when Gardner's column came out. The first prime factor was found in 1976.

Looking at the current status of the exponent 8191, one other prime factor is known, and the remaining cofactor is a PRP.
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