| HN Mirror

My reaction to the above link is "lolwut". The page talks about the fact that the digital sums ("horizontal addition") of the powers of 2 form a repeating sequence (1,2,4,8,7,5), and proceeds to call this "the MATHEMATICAL FINGERPRINT OF GOD".

I can sympathize with people who look at the whole beautiful structure of mathematics and see the touch of God, but the premise of that page seems to be that this particular phenomenon--by itself--is huge and revolutionary and supernatural. In fact, a student of modular arithmetic would see it as fairly trivial. Two observations suffice to explain it:

1. The digital sum of a decimal integer is just that integer mod 9. (This gives us, as a special case, the divisibility rule for 9: N is divisible by 9 iff its digits add to a multiple of 9.) To illustrate with five digits, the number N = abcde (base 10) is equal to:

  N = a * 10^4 + b * 10^3 + c * 10^2 + d * 10^1 + e

and since 10 ≡ 1 (mod 9), this reduces to

  N ≡ a * 1^4 + b * 1^3 + c * 1^2 + d * 1^1 + e (mod 9)
    ≡ a + b + c + d + e (mod 9).

If a+b+c+d+e < 10, then clearly that is the digital sum, and it's also N mod 9. If a+b+c+d+e ≥ 10, then we take the digital sum of that, which will again be the same as taking it mod 9. For a quick concrete example, the number 758 -> 7+5+8 = 20 -> 2+0 = 2, and 758 = 2 + 9 * 84, so it is 2 mod 9. Or we could say that 758 = 7 * 100 + 5 * 10 + 8 = 7 * 1 + 5 * 1 + 8 = 7+5+8, which is clearly the same number we got while taking the digital sum.

2. The powers of 2 mod 9 are periodic. (In general, the powers of N mod M are periodic, for any integers N and M.) Specifically, they are: 1, 2, 4, 8, 16 ≡ 7 (mod 9), 14 ≡ 5 (mod 9), 10 ≡ 1 (mod 9), and we are back at 1. This is exactly the "1,2,4,8,7,5" sequence that page talks about.

So, the above page talks about an elementary result of modular arithmetic as though it, by itself, were some logic-defying mystery. Seems pretty dishonest to me. Maybe it's a joke site. I saw a donation link, so maybe these guys are successful trolls.

I'm a bit curious as to the motives of the parent commenter. You seem to be a long-time member of HN (account is more than 2 years old). Was it a joke, or did you suspect something fishy but didn't know what it was and posted it here hoping someone would explain it for you, or did you actually take the author at his word, or what?