Abstract:

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 $a\not\equiv 1\bmod 12$, then $a$ is even. If $a$ is even and $3\nmid N$ then $m$ is also even. If $a\equiv 1\bmod 12$ then $3\mid N$ and $m$ is even. If $N$ is flat and 3-perfect and $3\nmid N$, then if $a\not\equiv 1\bmod 12$, $a$ is even. If $a\equiv 1\bmod 12$ 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.