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 of prime numbers p one, p two, ..., pn. it will be shown that at least one addi
tional prime number not in this list exists. let x be the product of all the pri
me numbers in the list. x equals p one times p two times ... pn. let q equals x 
plus one. then q is either prime or not. if q is prime, then there is at least o
ne more prime that is not in the list. if q is not prime, then some prime factor
 p divides q. if this factor p were on our list, then it would divide x, since x
 is the product of every number on the list. but p divides x plus one equals q. 
if p divides p and q, then p would have to divide the difference of the two numb
ers, which is x plus one minus x or just one. since no prime number divides one,
 p cannot be on the list. this means that at least one more prime number exists 
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