Show that there are infinitely many positive prime numbers.
Answer:
- Let us assume that there are a finite number of positive prime numbers namely, ^@p _1, \space p _2, \space p _3 \space ..... \space p _n^@, such that ^@p _1 < p _2 < p _3 \space ..... \space < p _n.^@
- Let ^@x^@ be any number such that,
^@x = 1 + \left(p _1 \times p _2 \times p _3 \times ..... \times p _n\right)^@
Observe that ^@\left(p _1 \times p _2 \times p _3 \times ..... \times p _n\right)^@ is divisible by each of ^@p _1, \space p _2, \space p _3 \space ..... \space p _n^@ but ^@x = 1 + \left(p _1 \times p _2 \times p _3 \times ..... \times p _n\right)^@ is not divisible by any of ^@p _1, \space p _2, \space p _3 \space ..... \space p _n^@. - Since ^@x^@ is not divisible by any of the prime numbers ^@p _1, \space p _2, \space p _3 \space ..... \space p _n^@, therefore, ^@x^@ is either a prime number or has prime divisors other than ^@p _1, \space p _2, \space p _3 \space ..... \space p _n^@.
This contradicts our assumption that there are a finite number of positive prime numbers. - Hence, there are infinitely many positive prime numbers.