| Prime numbers are more than any assigned multitude of prime numbers. | ||
| Let A, B, and C be the assigned prime numbers.
I say that there are more prime numbers than A, B, and C. | ||
| Take the least number DE measured by A, B, and C. Add the unit DF to DE.
Then EF is either prime or not. First, let it be prime. Then the prime numbers A, B, C, and EF have been found which are more than A, B, and C. |
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| Next, let EF not be prime. Therefore it is measured by some prime number. Let it be measured by the prime number G. | VII.31 | |
| I say that G is not the same with any of the numbers A, B, and C.
If possible, let it be so. Now A, B, and C measure DE, therefore G also measures DE. But it also measures EF. Therefore G, being a number, measures the remainder, the unit DF, which is absurd. Therefore G is not the same with any one of the numbers A, B, and C. And by hypothesis it is prime. Therefore the prime numbers A, B, C, and G have been found which are more than the assigned multitude of A, B, and C. | ||
| Therefore, prime numbers are more than any assigned multitude of prime numbers. | ||
| Q.E.D. | ||
Book IX Introduction - Proposition IX.19 - Proposition IX.21.