320 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1959 
fornia Institute of Technology has constructed seemingly simple nu- 
merical problems that a computer simply cannot cope with. In some 
cases the computer gets grossly inaccurate results; in others it cannot 
produce any answer at all. It is a challenge to numerical analysts 
to find ways to foresee this sort of trouble and then avoid it. 
PATTERNS IN PRIMES 
Computers have as yet made few direct contributions to pure 
mathematics except in the field of number theory. Here the results 
have been inconclusive but interesting. D. H. Lehmer of the Univer- 
sity of California has had a computer draw up a list of all the prime 
numbers less than 46,000,000. (A prime is a number that is exactly 
divisible only by itself or one—e.g. 2, 3, 17, 61, 1,021.) A study of 
the list confirms that prime numbers, at least up to 46,000,000 are dis- 
tributed among other whole numbers according to a “law” worked out 
theoretically about a century ago. The law states that the number 
of primes less than any given large number, X, is approximately 
equal to XY divided by the natural logarithm of X. (Actually, the 
approximation is consistently a little on the low side.) Lehmer’s list 
also tends to confirm conjectures about the distribution of twin 
primes—i.e., pairs of consecutive odd numbers both of which are 
primes, like 29 and 31, or 101 and 103. The number of twin primes 
less than XY is roughly equal to X divided by the square of the natural 
logarithm of X. 
Lehmer and H. S. Vandiver of the University of Texas have also 
used a computer to test a famous theorem that mathematicians the 
world over are still trying either to prove or disprove. Three hun- 
dred years ago the French mathematician Fermat stated that it is 
impossible to satisfy the following equation by substituting whole 
numbers (except zero) for all the letters if m is greater than 2: 
Lehmer and Vandiver have sought to find a single exception. If 
they could, the theorem would be disproved. Fortunately they have 
not had to test every conceivable combination of numbers; it is suffi- 
cient to try substituting all prime numbers for n. And there are 
further shortcuts. The number n, for example, must not divide any 
of a certain set of so-called “Bernoulli number”; otherwise it cannot 
satisfy the equation. (The Bernoulli numbers are irregular. The 
Ist is 1/6; the 3d, 1/30; the 11th, 691/2,730; the 13th, 7/6; the 
17th, 43,867/798; the 19th, 1,222,277/2,310. Numbers later in the 
series are enormous. ) 
Lehmer and Vandiver have tested the Fermat theorem for all prime 
n’s up to 4,000, but they seem to be coming to a dead end. The Ber- 
noulli numbers at this stage are nearly 10,000 digits long, and even 
