How do you figure out whether 999919 is a prime number or not using only paper-pen calculations in few minutes?

To test primality, you need to check whether the number is divisible by the prime numbers less than the integer part of its square root. I did it by noticing that 999,919 is almost 1,000,000 when I was determining what number would be close to its square root. The square root of 1,000,000 is 1,000. Then, since 999,919 is 1,000,000 - 81, it is one square number minus another: 1,0002 - 92. So, I know that 999,919 can be factored because of the Pythagorean theorem: a2 + b2 = c2, which can be rewritten as a2 = c2 - b2 = (c

• b)(c + b). By substituting 999,919 for a2, 9 for b, and 1000 for

c, I have 999,919 = (1000 - 9)(1000 + 9) = 991 x 1009. So, because 999,919 has factors other than 1 and itself, it cannot be a prime number.