If 2^p -1 is a prime, then the sum of powers (consisting of p elements) existing between 2^p-1 and 2^2(p-1), extremes included, is a perfect.

Example: p=3; 2^p -1 = 7;

then the sum of 2^2 + 2^3 + 2^4 =

= 4 + 8 + 16 (elements = 3 = p) = 28;

Example:p=5; 2^p -1 = 31;

then the sum of 2^4 + 2^5 + 2^6 + 2^7 + 2^8 =

= 16 + 32 + 64 + 128+ 256 (elements = 5 =p) = 496;

Example: p=7; 2^p -1= 127;

then the sum of 2^6 + 2^7 + ...+ 2^12

= 64 + 128 + ... + 4096 (element = 7 = p) = 8128.

Filippo Giordano

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