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| Title: | Mathematics at DEC |
|
| Moderator: | RUSURE::EDP |
|
| Created: | Mon Feb 03 1986 |
| Last Modified: | Fri Jun 06 1997 |
| Last Successful Update: | Fri Jun 06 1997 |
| Number of topics: | 2083 |
| Total number of notes: | 14613 |
365.0. "Min k with 2**k*n-1 composite" by R2ME2::GILBERT () Mon Oct 21 1985 18:34
(Carl Pomerance, from the 1983 Western Number Theory Conferences)
k
Let k = k(n) be the smallest non-negative integer such that 2 n - 1 is
composite. Prove that limit k(n)/n -> 0.
Note that k = 0 unless n = 1, 2, or one more than a prime. Moreover, if
p is a prime, having 2 as a primitive root, which does *not* divide n,
then k <= p-2, and there considerations modulo other primes; so large k
are hard to come by.
[ the above questions are rather hard, but Pomerance also asks: ]
What's the next biggest k after k(90) = 6 ?
| T.R | Title | User | Personal
Name | Date | Lines |
|---|
| 365.1 | | R2ME2::GILBERT | | Tue Oct 22 1985 00:29 | 9 |
| Up through 11 million, I've found:
k( 1122660) = 7
k( 2164230) = 7
k( 2329470) = 7
k(10257810) = 7
k(10309890) = 7
Note that, for k(n) >= 4, n must be a multiple of 30.
|
| 365.2 | | R2ME2::GILBERT | | Tue Oct 22 1985 16:54 | 39 |
| Here is my list of 'large' k (all those >= 7). There are no 9s (yet or ever?).
k(1122660) = 7
k(2164230) = 7
k(2329470) = 7
k(10257810) = 7
k(10309890) = 7
k(12314700) = 7
k(14030310) = 7
k(14145540) = 7
k(19099920) = 8
k(23103660) = 7
k(24176130) = 7
k(28843650) = 7
k(37088730) = 7
k(38199840) = 7
k(42389520) = 7
k(49160100) = 7
k(50785440) = 7
k(52554570) = 8
k(62800170) = 7
k(68672190) = 7
k(70370520) = 7
k(72592110) = 7
k(73091130) = 7
k(76852230) = 7
k(78620940) = 7
k(81991650) = 7
k(82556460) = 7
k(98810250) = 7
k(105109140) = 7
k(112874580) = 7
k(117720330) = 7
k(129918750) = 7
k(204474270) = 7
k(214914210) = 7
k(218029530) = 7
k(228067320) = 8
k(228111870) = 7
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| 365.3 | | R2ME2::GILBERT | | Wed Oct 23 1985 17:21 | 15 |
| And some more (including one with k = 9):
k(228111870) = 7
k(241034640) = 8
k(268277520) = 7
k(405419220) = 7
k(406484760) = 7
k(418244760) = 7
k(425284860) = 7
k(427827270) = 9
k(428162850) = 7
k(428230530) = 7
k(431189640) = 7
k(440430630) = 7
k(448474050) = 7
|