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-   -   Digit sums, of any value? (https://www.mersenneforum.org/showthread.php?t=26520)

 Chris65 2021-02-20 09:12

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

 retina 2021-02-20 09:39

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_sum[sup]n[/sup](x) = x mod 9

 Nick 2021-02-20 09:48

[QUOTE=Chris65;572072]
Is there a rule to explain it?[/QUOTE]
Look up the link between even perfect numbers and Mersenne primes.
Then examine it modulo 9, as retina suggests.

 R. Gerbicz 2021-02-20 09:53

[QUOTE=retina;572074]BTW: Repeated digit sum is just another way of computing the remainder when divided by 9.
[/QUOTE]

Repeated digit sum has a name: [url]https://en.wikipedia.org/wiki/Digital_root[/url] .

 sweety439 2021-02-21 15:23

[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)
[/CODE]

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