Some important Results in number theory

Suchir derived the following important results in number theory:

A. Last two digits of 3^(3^(3^(...........(3^3))).....) is always 87 as long as the number of 3s in the sequence of exponents is 3 or more.

B. Last two digits of 7^(7^(7^(...........7^7))).....) is always 43 as long as the number of 7s in the sequence of exponents is 2 or more.

C. Last two digits of 13^(13^(13^(...........(13^13))).....) is always 53 as long as the number of 13s in the sequence of exponents is 2 or more.

Some other important results:

1. Any natural number being a perfect square cannot have its sum of digits a number which leaves a remainder 2,3,5,6 or 8 when divided by 9. In other words the beejank of a square number can only be 1,4,7 or 9

2. The beejank of a perfect cube can only be 1,8 or 9.

3. The beejank of a perfect fourth power can only be 1,4,7 or 9.

4. The beejank of a perfect 6th power of a natural number can only be 1 or 9.

5. The cube of any number leaves a remainder 1,6 or 0 when divided by 7.

5. The numbers (x^2-y^2), (2xy) and (x^2+y^2) generates a Pythagorean triplet for any natural numbers x & y. When x&y are co-prime we get a primitive Pythagorean triplet.

5.1 The product of any Pythagorean triplet is always a multiple of 60.

6. Any Right Angled triangle with hypoteneuse being a prime of the form 4n+1 or a multiple of a prime of the form 4n+1 can have all all sides integral in length. (* This follows from the Fermat's Theorem i.e. any prime of the form 4n+1 can be expressed as a sum of two squares).