PRIME NUMBER, an integer indivisible by any number save itself and unity, that is, one which cannot be factored; such number are 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, etc. They occur very frequently among the lower numbers and begin to be scarce among the higher thousands. The determination of the prime numbers has remained a mere matter of rejecting factorable numbers; this is done in a quasi-mechanical way by the table called Eratos thenes' sieve, upon which all the numbers to be tested are written at equal intervals and covered in succession by gridiron-like slips of paper, covering first all multiples of 2 (the even num bers may better be omitted at the start), then all multiples of 3 and so on. But even with the modern theory of numbers no rule has been laid down for the periodicity of the oc currence of the prime numbers. Gauss dis covered in 1810 that the number of primes less 2 dx than x was approximately i Chebicheff gave a formula for the etermination of the number of prime numbers between any two primes; in a monograph dated 1859, Riemann gave an even more exact formula and a very complicated one, also dependent on the integral used by Gauss. The theory of numbers has
also shown that every prime number is (in general in more than one way) a sum of four squares,>) but this is equally true of other num bers, inasmuch as (1) every factorable number is the product of two prime numbers and (2) the sum .of four squares into the sum of four squares is always a sum of four squares. There is a theorem of Goldbach to the effect that ((every even number is the sum of two prime numbers?' Factor tables have been made for all numbers between 1 and 10,000,000 in the attempt to codify the prime numbers; by refer ence to them it may be instantly discovered whether a given number is prime or not.