Immaculate perfection

— I am so lucky. I was born on April 4th 1944, that’s 4/4/44. If you add that up, it comes to 16, one six. One plus six is seven: luckiest number of all.
— You know your math.
— It’s more than math, Mike, it’s… immaculate perfection.

The value of 7 for the the digital root indeed seems preferable, at least on this blog. But looking at the dates formed by repeated digits is too anthropocentric (not to mention xkcd1179). Besides, mathematics already has a concept of perfect numbers: being equal to the sum of proper divisors.

So, are there any perfect numbers with digital root equal to 7, equivalently {n\equiv 7\bmod 9}? For even perfect numbers, one can expect that the Euclid-Euler theorem {n = 2^{p-1} (2^p-1)} will help with the answer, but even so, the solution turns out to be surprisingly simple.

A lucky coincidence is that {2^6\equiv 1\bmod 9} ({64 = 1 + 7\cdot 9}). Since all primes greater than 3 have remainder 1 or 5 mod 6, there are only a few cases to check:

  • {p\equiv 1\bmod 6}, hence {2^{p-1}\equiv 2^{1-1} \equiv 1 \bmod 9} and {2^p-1\equiv 2^1-1 \equiv 1\bmod 9}. In this case {n\equiv 1\cdot 1 \equiv 1 \bmod 9}
  • {p\equiv 5\bmod 6}, hence {2^{p-1}\equiv 2^{5-1} \equiv 7 \bmod 9} and {2^p-1\equiv 2^5-1 \equiv 4\bmod 9}. In this case {n\equiv 7\cdot 4 \equiv 1 \bmod 9}
  • Exceptional cases p=2, 3 correspond to n = 6, 28. The latter is congruent to 1 mod 9, matching the preceding cases.

So, every even perfect number except 6 is congruent to 1 mod 9. Should a perfect number be congruent to 7 mod 9, it has to be odd. An odd perfect number, should it exist, must be either divisible by 9 (not good for us) or congruent to 1 mod 12. (Reference: On the form of an odd perfect number). The condition {n\equiv 1 \bmod 12} is consistent with {n\equiv 7 \bmod 9}: for example, 25 satisfies both. Come to think of it, the sum of proper divisors of 25 is… not 25. But there might still be a 7 mod 9 perfect number out there.

What numbers k have the property that the sequence {n mod k : n is perfect} stabilizes? Assuming odd perfect numbers do not exist, every power of 2 has this property. So do 3 and 9 by the above (and therefore, their products with powers of 2). I don’t know any other such k, in particular n mod 27 does not appear to stabilize. Of course, it is impossible to prove any negative result here since we don’t know if there are infinitely many perfect numbers.