1. x is the greatest prime

2. Form the product of all primes less than or equal to x, and add 1 to the product. This yields a new number y, where y=

[2x3x5x7x11x...x[x]] +1

3. If y is itself prime, then x is not the greatest prime, for y is obviously greater than x

4. If y is composite [not prime], then again, x is not the greatest prime. For if y is composite, it must have a prime divisor z; and z must be different from each of the prime numbers [2,3,5,7,11,...,x] smaller than or equal to x; hence z must be a prime greater than x

5. But y is either prime or composite

6. Hence x is not the greatest prime

7. There is no greatest prime