mersenneforum.org Digit sums, of any value?
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 2021-02-20, 09:12 #1 Chris65   Feb 2021 1 Posts Digit sums, of any value? Hello, I have not worked this out mathematically, I am simply studying primes in my spare time, but now as I look at the table of all of the Perfect numbers found it looks as though the digit sums of the digit sums add up to 10. Except for 6 and 28 this holds for the first 8 Perfect numbers: 28 the first digit sum = 2+8=10 496=4+9+6=19, 1+9=10 8128=8+1+2+8=19, 1+9=10 33550336=3+3+5+5+0+3+3+6=28, 2+8=10 Next digit sum=64, 6+4=10 Next digit sum=55, 5+5=10 Next digit sum=73, 7+3=10 Is there a rule to explain it? Could it be used to find more Perfect numbers? /Sincerely, Chris65
 2021-02-20, 09:39 #2 retina Undefined     "The unspeakable one" Jun 2006 My evil lair 11·569 Posts And the digit sum of 10 is 1. BTW: Repeated digit sum is just another way of computing the remainder when divided by 9. Or more succinctly: digit_sumn(x) = x mod 9
2021-02-20, 09:48   #3
Nick

Dec 2012
The Netherlands

6B616 Posts

Quote:
 Originally Posted by Chris65 Is there a rule to explain it?
Look up the link between even perfect numbers and Mersenne primes.
Then examine it modulo 9, as retina suggests.

2021-02-20, 09:53   #4
R. Gerbicz

"Robert Gerbicz"
Oct 2005
Hungary

27218 Posts

Quote:
 Originally Posted by retina BTW: Repeated digit sum is just another way of computing the remainder when divided by 9.
Repeated digit sum has a name: https://en.wikipedia.org/wiki/Digital_root .

 2021-02-21, 15:23 #5 sweety439     "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 52×112 Posts Code: n possible values for even perfect numbers mod n 1 0 2 0 3 0 (only for 6), 1 4 0, 2 (only for 6) 5 1, 3 6 0 (only for 6), 4 7 0 (only for 28), 1, 6 8 0, 4 (only for 28), 6 (only for 6) 9 1, 6 (only for 6) 10 6, 8 11 1, 4, 6, 10 12 4, 6 (only for 6) 13 1, 2, 3, 6 (only for 6), 8 14 0 (only for 28), 6, 8 15 1, 6 (only for 6), 13 16 0, 6 (only for 6), 12 (only for 28) 17 1, 2, 3, 6 (only for 6), 11 18 6 (only for 6), 10 19 1, 2, 3, 6 (only for 6), 7, 9 (only for 28), 10, 15 20 6 (only for 6), 8, 16 21 1, 6 (only for 6), 7 (only for 28), 13 22 4, 6, 10, 12 23 1, 3, 5, 6, 9, 13, 15, 20 24 4 (only for 28), 6 (only for 6), 16 25 1, 3, 6, 11, 16, 21 (only for 496) 26 2, 6 (only for 6), 8, 14, 16 27 1, 6 (only for 6), 10 28 0 (only for 28), 6 (only for 6), 8, 20 29 1, 3, 4, 6, 7, 8, 16, 18, 26, 28 30 6 (only for 6), 16, 28 31 0 (only for 496), 1, 6, 27, 28 32 0, 6 (only for 6), 16 (only for 496), 28 (only for 28)

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