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Old 2008-10-12, 20:41   #1
davar55
 
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Default Prime-Empty Century

The first prime-empty decade (x0 thru x9 composite) occurs for
x=20, since 200 to 209 are composite.

What is the first prime-empty century (x00 thru x99 composite)?

What about the first prime-empty millenium?
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Old 2008-10-12, 22:18   #2
cheesehead
 
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Could we adopt more optimistic terminology by referring to "compositeful [to coin a word] centuries", or simply "composite centuries"?

Or at least "prime-free centuries" rather than "prime-empty centuries"?
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Old 2008-10-12, 22:52   #3
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Quote:
Originally Posted by davar55 View Post
What is the first prime-empty century (x00 thru x99 composite)?
x=16719
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Old 2008-10-12, 23:18   #4
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Quote:
Originally Posted by Kevin View Post
x=16719
off-by-1

There are no primes between 1671781 and 1671907.

I find no composite millenium below p[sub]1000000[/sub] = 15485863.

Getting those results required using only the search-and-replace and duplicate-line-deleting functions of a text editor, plus visually comparing a few line numbers to line content, on a file listing the first million primes in order. I need to generate some larger lists.

Last fiddled with by cheesehead on 2008-10-12 at 23:40
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Old 2008-10-13, 00:00   #5
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Blame Maple for incrementing the counter on a for-while loop even when the while condition isn't met (and me for screwing up the double check because I used the wrong letter).
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Old 2008-10-13, 00:17   #6
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[I]Classic[/I] off-by-1 example!
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Old 2008-10-13, 04:25   #7
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No prime-free centuries x000-x999 up to x=27414773
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Old 2008-10-13, 08:41   #8
fivemack
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http://www.trnicely.net/gaps/gaplist.html

suggests that the first prime-free millennium will be somewhere around 10^16, so naive search techniques are unlikely to be very useful.

1100 CNC Be.Nyman 2001 29.02 17 28907866922785967

indicates that 28907866922786xxx is a prime-free millennium, and, if you trust that the list of kilogaps below 5e16 is complete, it's probably the first one.
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Old 2008-10-13, 08:43   #9
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Quote:
Originally Posted by Kevin View Post
No prime-free centuries x000-x999 up to x=27414773
A millennium lasts from x001 to (x+1)000. Otherwise the first millennium would have only 999 years. (There is no january 0th and no year 0.) Centuries last from x01 to (x+1)00 for the same reason.
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Old 2008-10-13, 09:09   #10
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Quote:
Originally Posted by S00113 View Post
A millennium lasts from x001 to (x+1)000. Otherwise the first millennium would have only 999 years.
I recall debating this in a Y2k forum during a year whose Gregorian calendar numeric designation was 199something. Both sides can be argued well, depending on circumstances and definitions.

Quote:
(There is no january 0th and no year 0.)
But here we're dicussing plain integer numbers, not measures of time, and, unlike many folks 2000 years ago, we all pretty much understand the concept of the number 0 (zero). Here, the "centuries" and "millenia" are only analogies (span of 100, span of 1000) referring to the original time-related meanings of the words, and the integer 0, unlike either January 0 or year 0, does exist.

- - -

Edit: However, there's that joke about imaginary numbers and one-half piece of chalk, to illustrate that the number one-half may not exist in certain contexts ...

Last fiddled with by cheesehead on 2008-10-13 at 10:04
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Old 2008-10-13, 09:14   #11
cheesehead
 
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Quote:
Originally Posted by fivemack View Post
http://www.trnicely.net/gaps/gaplist.html

suggests that the first prime-free millennium will be somewhere around 10^16, so naive search techniques are unlikely to be very useful.
"Naive"? I thought my use of that text editor's functions to accomplish a mathematical goal was at least somewhat sophisticated. It required the simultaneous (or, at least, overlapping) consideration of text as both character strings and positive integers, after all. ;-)

However, I do thank you for that estimate. Saves me a lot of disk space. :-)

Quote:
1100 CNC Be.Nyman 2001 29.02 17 28907866922785967
There's

1050 CFC Be.Nyman 2000 28.25 17 13893290219203981

Last fiddled with by cheesehead on 2008-10-13 at 09:33
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