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. Ifa ≡ 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.
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机译：我们说一个数字是平坦的，如果它可以被写成2乘以一个奇方免费数字的非平凡幂。幂是“指数”，奇数个素数是“长度”。令N为平坦且4完美的，指数为a，长度为m。如果a 1 mod 12，则a是偶数。如果a为偶数且3∤N，则m也为偶数。如果a 1 mod 12，那么3 | N和m是偶数。如果N是平坦的且3完美且3 3 N，则如果a≡1 mod 12，则a是偶数。如果a 1 mod 12，则m为奇数。如果N是平坦且3或4完美的，那么它可以被至少一个Mersenne素数整除，但并非所有奇数素数因子都是Mersenne。我们还给出了一些条件，可以将任意偶数为4的完美数除以3。
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