G.350 prime numbers more

IX.20

Prime numbers are more than any assigned multitude of prime numbers.

prime_numbers_more(A, B, C)

given A, B, C be the assigned prime numbers

prove there are more prime numbers than A, B, C; G is not the same with any of the numbers A, B, C

given A, B, C be the assigned prime numbers

construct Let the least number measured by A, B, C be taken

assert Let it be / D E /

construct Let the unit / D F / be added to / D E /.

assert Then / E F / is either prime or not

assert First, let it be prime

assert Then the prime numbers A, B, C, / E F / have been found which are more than A, B, C

assert Next, let / E F / not be prime

∵ composite_number_measuredassert it is measured by some prime number

construct Let it be measured by the prime number G.

assert If possible, let it be so

assert Now A, B, C measure / D E /

assert G also will measure / D E /

assert It also measures / E F /

assert G, being a number, will measure the remainder, the unit / D F / ⇒⇐

assert G is not the same with any one of the numbers A, B, C

assert By hypothesis it is prime

∴ the prime numbers A, B, C, G have been found which are more than the assigned multitude of A, B, C

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