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Sunday, July 13, 2008

Sunday July 13, 2008

Filed under: General — Tags: — m759 @ 10:23 pm
The Drunkard’s Walk
is the title of a recent
book by Leonard Mlodinow:

http://www.log24.com/log/pix08/080713-DrunkardsWalk.jpg
 
Cover of British edition


“Leonard Mlodinow has had, to speak informally, a pretty random career….

A far more sober instance of randomness, however, underpins his new book, The Drunkard’s Walk. And it’s not hard to see it as a sort of Rosebud, explaining why the author finds unpredictability so compelling.”

Another sort of Rosebud–
C. P. Snow on G. H. Hardy:

“… A Mathematician’s Apology is, if read with the textual attention it deserves, a book of haunting sadness. Yes, it is witty and sharp with intellectual high spirits: yes, the crystalline clarity and candour are still there: yes, it is the testament of a creative artist. But it is also, in an understated stoical fashion, a passionate lament for creative powers that used to be and that will never come again.”

Perhaps in the afterlife Hardy, an expert on the theory of numbers, does again enjoy such powers. If so, his comments on the following would be of interest:

New York Lottery today:
Mid-day 006
(the first perfect number)
Evening 568
(an apparently random number)

Hardy, when taken to church as a child, passed the time by factorizing hymn numbers. This suggests we note that 568 equals 8 times 71. A check of Wikipedia on the prime number 71 reveals that it is related to 568 in another way: 568 is is the sum of the primes less than 71–

2 + 3 + 5 + 7 + 11 +
13 + 17 + 19 + 23 +
29 + 31 + 37 + 41 +
43 + 47 + 53 + 59 +
61 + 67 = 568
Clearly it is false that the sum of the primes less than a prime p is, in general, a multiple of p, since (2 + 3 + 5) is not a multiple of 7. The sum of primes less than an integer x is, however, of some interest.

See The On-Line Encyclopedia
of Integer Sequences,

A046731, Sum of primes < 10^n, as well as
A006880, Number of primes < 10^n.

According to an amateur* mathematician named Cino Hilliard, “a very important relationship exists” between the sum of primes less than x and the prime counting function Pi(x) which is the number of primes less than x

(Sum of primes less than x) ~ Pi(x^2).

Whether this apparent relationship is, in fact, “very important,” or merely a straightforward consequence of other number-theoretical facts, is not obvious (to those of us not expert in number theory) from Google searches. Perhaps Hardy can clear this question up for those who will, by luck or grace, meet him in the next world.

* For some background, see a profile and user group messages here and here and here.

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