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Sunday, March 27, 2016

The Contradiction of Primes


Primes been on my mind lately, so, for Sunday reflection:
"There are two facts about the distribution of prime numbers which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts.
"The first is that despite their simple definition and role as the building blocks of the natural numbers, the prime numbers... grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout.
"The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behaviour, and that they obey these laws with almost military precision."
-- D. Zagier  in "The First 50 Million Prime Numbers" The Mathematical Intelligencer  (1977)

...and:
"One of the remarkable aspects of the distribution of prime numbers is their tendency to exhibit global regularity and local irregularity. The prime numbers behave like the 'ideal gases' which physicists are so fond of. Considered from an external point of view, the distribution is -- in broad terms -- deterministic, but as soon as we try to describe the situation at a given point, statistical fluctuations occur as in a game of chance where it is known that on average the heads will match the tail but where, at any one moment, the next throw cannot be predicted. Prime numbers try to occupy all the room available (meaning that they behave as randomly as possible), given that they need to be compatible with the drastic constraint imposed on them, namely to generate the ultra-regular sequence of integers."
-- G. Tenenbaum and M. Mendèfs France, in "The Prime Numbers and Their Distribution" (AMS, 2000)


[These come from a great collection of quotes about prime numbers from a page at Matthew Watkins' website:
http://empslocal.ex.ac.uk/people/staff/mrwatkin//isoc/quotes.htm ]


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