Perfect Numbers
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Mersenne Number Formula: (2^P -1 )
The Oldest Unsolved Problem in Math
Are there any odd Perfect Numbers???????
Veritasium
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* Youtube Video *
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Sigma(N) = Summ of all its Divisors (not Factors) including itself
A Perfect Number is a number = Summ of all of its Divisors
Prime Divisors= = Sum of all its Divisors = Perfect Numbers
1x2x3 = 6 = 1+2+3 = 1+2+3 =4x1 + 2
1x2x4x7 = 28 = 1+2+4+7+14 = 1+2+3+4+5+6+7 =8x3 + 4
1x2x4x31 = 248 = 1+2+4+8+16+31+62+124 = 1+ 2+ 3+ 4+ 5+ 6+ 7+ 8+ 9+10+11
21+20+19+18+17+16+15+14+13+12 (doesn't work)
33th
only if (2^P)-1 is prime # of digits
P Perfect in ((2^P) -1 ) x 2^(P -1) (base 10) N
Number binary
2 6 110 4-1 x 2^1 = 3 x 2 1 1
3 28 11100 8 - 1 x 2^2 = 7 x 4 2 2
5 496 111110000 3 3
7 8128 1111111000000 4 4
13 33550336 8 5
17 8589869056 10 6
19 137438691328 12 7
31 2305843008139952128 2147483647 x 2^30 = 19 8
2282 ((2^2281)-1) x 2^(2281-1) 687 17
Some Mersenne Numbers are Prime Numbers
All Mersenne Primes are Perfect Numbers
(2^P -1 ) x 2^(P -1) Mersenne Primes x 2^(P-1)
2 to 77232917 - 1 (50th Mersenne Prime) in 2017 50
2 to the 82589933 -1 (51st Mersenne Prime) in 2018 49,724,095 51
2 to the 100000000-1 none as of 2020LDec
unknown (52nd Mersenne Prime) as of 2023DApr03
on 2024DApr03
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