Show that there are infinitely many positive prime numbers.


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Answer:


Step by Step Explanation:
  1. 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.^@
  2. 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^@.
  3. 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.
  4. Hence, there are infinitely many positive prime numbers.

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