Mathematical_Recreations-Kraitchik-2e

78 Mathematical Recreations
The last digits may be either
3 740 081 787 109 376
6 259 918 212 890 625.
9. PRIME NUMBERS
As one goes out in the sequence of natural numbers, prime
numbers become more and more rare. For example, there
are 26 prime numbers between ° and 100, only 21 between
100 and 200, and no more than 4 between IOU and 1012 + 100.
In spite of the rarity of the primes, they exist among numbers
no matter how large. We can never name the largest
prime number, but only the largest known prime number,
that is to say, the largest number whose primality has been
proved.
Euler proved that 231 -
1 is a prime number, and this was
the largest known prime number for a century, until Landry,
Lucas, Lehmer, Powers and other authors of our time broke
Euler's record.
Now we know many numbers greater than Euler's number.
Sphinx at one time listed all known isolated prime numbers
greater than 1012, and all prime numbers between 1012 -
10,000 and 1012 + 10,000. This catalogue contains nine nUJ'llbers
greater than 1019, one of which was shown by Lehmer
to be incorrect. The other eight are:
The factor of (3105 _2105 ), (296 + 1) -;- (232 + 1), (1023
-1) -;- 9,5.275 + 1, 289 -1, (1031 + 1) -;- 11, 2107 -1,
2127 -1.
Moreover Miller and Wheller proved that k.(2127 -1)
+ 1 is a prime for k = 114,124,388,408,498,696,738,744,
780,934 and 978, as well as 180.(2127 _1)2 + 1, and Mr.
Ferrier proved that (2148 + 1) -;- 17 is prime.
Mr. R. M. Robinson proved with the electronic machine
the primality of 2n -1 for n = 521,607,1279,2203,2281.
Each of these numbers has its history, and the proof of its