We say a number is flat if it can be written as a non-trivial power of 2 times an odd
squarefree number. The power is the “exponent” and the number of odd primes the
“length”. Let N be flat and 4-perfect with exponent a and length m. If a ≢ 1 mod 12,
then a is even. If a is even and 3 ∤ N then m is also even. If a ≡ 1 mod 12 then 3 | N
and m is even. If N is flat and 3-perfect and 3 ∤ N, then if a a ≡ 1 mod 12, a is even. If
a ≡ 1 mod 12 then m is odd. If N is flat and 3 or 4-perfect then it is divisible by at
least one Mersenne prime, but not all odd prime divisors are Mersenne. We also give
some conditions for the divisibility by 3 of an arbitrary even 4-perfect number.
