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Prime Numbers: Digits
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Earlier this year it was announced that the world's largest prime had been found. The Great Internet Mersenne Prime Search has found the number 2 raised to the power 25964951 minus 1 to be prime... establishing the record. This record prime has more than 7.8 million digits! Mersenne primes are of the form 2 raised to a power minus 1 and are named after the French monk Marin Mersenne, who in the early 17th century published a list of these primes. The first few are 3, 7 and 31, with the latest record prime only the 42nd Mersenne prime.
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You determine prime numbers by the process of elimination. The way you determine if the number n is prime is to see if you can divide it by any number between 1 and n/2. Proving that a given 14 digit number is prime involves several trillion operations.
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The Electronic Frontier Foundation (EFF) has offered a US$100,000 prize to the first discoverers of a prime with at least 10 million digits. They ... offer $150,000 for 100 million digits, and $250,000 for 1 billion digits. In 2000 they paid out $50,000 for 1 million digits.
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This is an example of a prime circle. Two adjacent numbers, including the last number and the first number, sum to a prime. In this particular case all the numbers are 3 digits. This circle is of length ninety, and is part of a 200 length prime circle found by Charles Ashbacher, JRM 26:1, 1994, p 63.
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In practice, though, one usually wants to check whether a given number is prime, rather than generate a list of primes. Further, it is often satisfactory to know the answer with a high probability. It is possible to quickly check whether a given large number (say, up to a few thousand digits) is prime using probabilistic primality tests. These typically pick a random number called a "witness" and check some formula involving the witness and the potential prime N. After several iterations, they declare N to be "definitely composite" or "probably prime". Some of these tests are not perfect: there may be some composite numbers, called pseudoprimes for the respective test, that will be declared "probably prime" no matter what witness is chosen. However, the most popular probabilistic tests do not suffer from this drawback.
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If the digits of the prime, reading left to right, steadily increase to a maximum value, and then steadily decrease, they are called peak primes. Valley primes are just the opposite. There are a total of 10 peak and 20 valley primes. 345676543 is unique because of the five consecutive digits.
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