How the twin primes to cause the error in the processor Pentium Intel?
Whitin the group of integers, prime numbers are in a way thought of as atoms, since all integers can be expressed as a product of prime numbers (for example, 30=2x3x5), just as molecules are made up of separate atoms.
The theory of prime numbers continues to be shrouded in mystery and still holds many secrets.
Taking the first 100 numbers we count 25 primes; between 1001 and 1100 there are only 16; and between the numbers 100,001 and 100100 there are a mere six.
Prime numbers become increasingly sparse. In other words, the average distance between two consecutive primes becomes increasingly large.
Around the turn of the 19th century, the Frenchman Adrien-Marie Legendre and the German Carl Friedrich Gauss studied the distribution of primes. Based on their investigations they conjectured the space between a prime P and the next bigger prime would on average, be as big as the natural logarithm of P.
Sometimes the gaps are much larger, sometimes much smaller. There are even arbitrarily long intervals in which no primes occur whatsoever. The smallest gap. On the other hand, are two, since there is at least one even number between any two primes.
Primes that are separated from each other by a gap of only two –for instance, 11 and 13, or 197 and 199- are called twin primes.
There are also prime cousins, which are primes separated from each other by four nonprime numbers. Primes that are separated from each other by six nonprime numbers are called, sexy primes.
Much less is known about twin primes than about regular primes. What is certain is that they are fairly rare.
Among the first million integers there are only 8169 twin prime pairs. The largest twin primes so far discovered have over 50,000 digits. But much is unknown.
Nobody knows whether an infinite number of twin prime pairs exist, or whether after one particular twin prime pair there are no larger ones.
Working on the theory of twin primes, Thomas Nicely from Virginia, in the 1990s. He was running through all integers up to 4 quadrillion.
The algorithm required the computation of the banal expression X times (1/X). But to his shock, when inserting certain numbers into this formula, he received not the value 1 but an incorrect result.
On October 30, 1994, his computer consistently produced erroneous results when calculating the above equation with numbers ranging between 824,633,702,418 and 824,633,702,449. Through his research on twin primes, Thomas Nicely had hit on the notorious Pentium bug.
The error in the processor cost Intel, the manufacturer, and $500 million in compensations.
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