The Twelve Days of Christmas: Day 6

Jeremy's Status Message Proudly Presents The Twelve Days of Christmas:

On the sixth day of Christmas, an evil number theorist said to me,

6 is the smallest perfect number, because it is equal to the sum of its proper positive divisors: 1, 2, and 3. Can you find the next even perfect number? Can you find an odd perfect number?

5 is a prime number whose binary representation, "101" is palindromic, meaning its digits read the same backward as forward. What is the largest known prime number that is palindromic in binary?

4 is an even integer that can be written as the sum of two primes, namely 2 and 2. Can you find a greater even integer that cannot?

3 is a triangular number, because it belongs to the sequence (1, 3, 6, 10, 15, 21, ...). The number 3 can be represented as a sum of three triangular numbers: 3 = 1 + 1 + 1. Can you find a greater integer that cannot?

2 is a value for n such that there exist nonzero integers x, y, and z where x^n + y^n = z^n. For example, 3^2 + 4^2 = 5^2. Can you find a value of n greater than 2 for which this is also true?

1 is the difference between 2 and 3. There also exist powers of 2 and 3 such that their difference is also 1, namely 2^3=8 and 3^2=9. Can you find another pair of consecutive integers who have powers that are also consecutive?

Explanation: I love number theory because it is primarily concerned with integers, which seem quite simple, and yet can be extremely complex. These are just a few examples. I will post the solutions tomorrow.