Perfect numbers
Mark Brader
Martin Gardner gave the last digit of each of the perfect numbers then
known. Of course, only even perfect numbers are known, and it is known
that all even perfect numbers are of the form (2^(n-1))*(2^n -1), where
2^n -1 is prime (called a Mersenne prime) and therefore n is prime also*.
From this it is easy to show that the last digit of an even perfect
number must be 8** if n is congruent to 3 modulo 4, and 6 otherwise.
Now, the observed list of last digits given in the article is:
6 8, 6 8; 6 6 8 8, 6 6 8 8; 6 8 8 8, 6 6 6 8; 6 6 6 6.
I have inserted punctuation marks to emphasize what Gardner calls the
"infuriating hints of order" in the sequence. Now, since this article
appeared, I know that several new Mersenne primes have been discovered,
and each one corresponds to a perfect number. Does anyone on the net
have easy access to a list of the numbers? It would be interesting to
see if they continue to exhibit anything resembling a pattern.
Note: I am not suggesting that I believe this is anything more than
coincidence, just like the 18281828 in the decimal expansion of e, and
the 999999 in that of pi. It's just interesting because it's pretty. I think.
Mark Brader
*If n was composite, say n=a*b, 2^n-1 would be equal to 2^a-1 times
2^(n-a) + 2^(n-2*a) + ... + 1, and therefore composite.
**In fact, the last two digits must be 28.