### 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.

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