April 18th, 2006

'Twixt Sleep and Wake

New phrases in my vocabulary: consolidated sleep and segmented sleep. I first heard about this from a friend, and did a little online research ... it seems that in many pre-industrial societies, it has always been the norm to sleep in two shifts, not the eight-hour stretch we aspire to in the electrified world. These two sleep periods were called "first sleep" and "second sleep" in the English language, and were often separated by a period of quiet contemplation, lovemaking, creativity, or dream interpretation.

Wikipedia:
Segmented sleep or divided sleep are modern Western terms for a sleep pattern found in medieval Europe and many modern non-industrial societies, where the night's sleep is evenly divided by a few hours of wakefulness.

The human Circadian rhythm controls a sleep-wake cycle of wakefulness during the day and sleep at night. Superposed on this basic rhythm is a secondary one of light sleep in the early afternoon (see nap and siesta) and quiet wakefulness in the early morning.
The two periods of night sleep were called first sleep (occasionally dead sleep) and second sleep (or morning sleep) in medieval England. First and second sleep are also the terms in the Romance languages, as well as the Tiv of Nigeria. There is no common word in English for the period of wakefulness between, apart from paraphrases such as first waking or when one wakes from his first sleep and the generic watch (in its old meaning of being awake). In French an equivalent generic term is dorveille ("twixt sleepe and wake").

There is evidence from sleep research that this period of nighttime wakefulness, combined with a midday nap, result in greater alertness than a single sleep-wake cycle. ...

American Historical Review article:
Until the close of the early modern era, Western Europeans on most evenings experienced two major intervals of sleep bridged by up to an hour or more of quiet wakefulness. In the absence of fuller descriptions, fragments in several languages that I have surveyed survive in sources ranging from depositions and diaries to imaginative literature. From these shards of information, we can piece together the essential features of this puzzling pattern of repose. The initial interval of slumber was usually referred to as "first sleep," or, less often, "first nap" or "dead sleep."65 In French, the term was "premier sommeil" or "premier somme,"66 in Italian, "primo sonno" or "primo sono,"67 and in Latin, "primo somno" or "concubia nocte."68 The intervening period of consciousness—what Stevenson poetically labeled a "nightly resurrection"—bore no name, other than the generic term "watch" or "watching" to indicate a period of wakefulness that stemmed, according to the Oxford English Dictionary, "from disinclination or incapacity for sleep." Two contrasting texts refer to the time of "first waking."69 The succeeding interval of slumber was called "second" or "morning" sleep.70 Both phases lasted roughly the same length of time, with individuals waking sometime after midnight before ultimately falling back to sleep.

Fibonacci Primes

You know that prime numbers are cool. And you know that Fibonacci numbers are way cool. But did you know just how cool Fibonacci primes are?

Fibonacci numbers, you know, are numbers in the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 ... where each successive number is the sum of the two before it. And primes can't be divided by any number except themselves. Fibonacci primes, as I'm sure you've already guessed, are Fibonacci numbers that are also prime.

But wait! Here's the super cool part. With the exception of the number 3, every known Fibonacci prime occupies a prime-numbered place in the sequence of Fibonacci numbers!!! No, really, I'm not making this up. Look, the first few Fibonacci primes are {2, 3, 5, 13, 89, 233, 1597 ...}. These are numbered third, fourth (there's that solitary exception), fifth, seventh, eleventh, and 13th on the Fibonacci sequence. The next number, 1597, comes in at 17th. And the pattern holds for as far as mathematicians are able to test it. Here's the Wolfram Research article on Fibonacci Primes:
The first few proven prime Fibonacci numbers are F(n) are 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, ... (Sloane's A005478), which occur for n=3, 4, 5, 7, 11, 13, 17, 23, 29, 43, 47, 83, 131, 137, 359, 431, 433, 449, 509, 569, 571, 2971, 4723, 5387, 9311, 9677, 14431, 25561, 30757, 35999, 37511, 50833, 81839, 104911, 130021, 148091, 201107, 397379, ... (Sloane's A001605; Dubner and Keller 1999), where the Fibonacci numbers with indices 104911 (B. de Water), 130021 (D. Fox), 148091 (T. D. Noe) and 201107, 397379, 433781, 590041, 593689, and 604711 (H. Lifchitz) are probable primes (Caldwell).

The article reminds us that "the converse is not true" and some numbers holding prime spots on the Fibonacci chain aren't prime themselves (the 19th Fibonacci number, 4181, isn't prime.)

But still. Cool, or what?

Well, maybe only for number geeks.
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