- #1
alexmahone
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Show that $n\equiv 1 \pmod 9$ for all even perfect numbers $n>6$.
Euge said:Hi Alexmahone,
Please check back to make sure your statement is the intended one, because as it stands it is false. For instance, $4^2 = 16$ is not congruent to $1\pmod{9}$ and $12^2 = 144 = 16\cdot 9$ is not congruent to $1\pmod{9}$.
An even perfect number is a positive integer that is equal to the sum of its positive divisors (excluding itself). In other words, the sum of all the divisors of an even perfect number, including 1 but excluding the number itself, is equal to the number.
This statement means that for any even perfect number $n$, when divided by 9, the remainder will always be 1. This has been observed to be true for all even perfect numbers greater than 6.
This relationship between even perfect numbers and 9 can help in the search for new even perfect numbers. It also provides a necessary condition for a number to be considered as a potential even perfect number.
No, it is not possible. This statement is a necessary condition for a number to be an even perfect number. If a number does not satisfy this condition, it cannot be an even perfect number.
The study of even perfect numbers is an ongoing area of research in mathematics. While there have been many significant discoveries and advancements made, there is still much to be explored and understood about these numbers and their properties.